Equations

Key Terms

What an Equation Is

Solutions and Solution Sets

Types of Equations by Solution Behavior

Absolute Value Equations

Equations Beyond Algebra

Summary of Algebraic Equation Types

Common Pitfalls

Equations FAQ

The Language of Mathematical Conditions

An equation is a statement that two mathematical expressions have the same value. Unlike an expression, which simply represents a quantity, an equation makes a claim — one that may be true, false, or true only under certain conditions. Determining when and whether that claim holds is the central problem of algebra, and nearly every branch of mathematics defines its own family of equations. This section treats equations as a subject in their own right: what they are, how they behave, and how the algebraic types are classified and solved.

Key Terms

Equations and Expressions

Equation— a statement that two expressions are equal, joined by the

=

sign

Expression— a mathematical phrase combining numbers, variables, and operations, with no equality sign

Variable— a symbol (typically

x

y

z

) representing an unknown or changing quantity

Solutions

Solution— a value of the variable that makes both sides of an equation equal

Solution set— the collection of all solutions to an equation

Extraneous solution— a value that emerges during solving but does not satisfy the original equation

Equation Classification

Conditional equation— true only for specific values of the variable

Identity— true for all permissible values of the variable

Contradiction— true for no value of the variable

Equivalent equations— equations that share the same solution set

Algebraic equation— built from variables, constants, and the operations

+

-

\times

\div

, and integer powers

Degree of an equation— the highest power of the variable after simplification

Standard form— all terms on one side, ordered by descending powers, equal to zero

Supporting Concepts

Coefficient— the numerical factor multiplying a variable or power of a variable

Discriminant— the quantity

\Delta = b^2 - 4ac

in a quadratic equation, determining the number and type of solutions

Domain restriction— a value excluded because it causes division by zero or undefined operations

Absolute value— the distance of a number from zero on the number line, always non-negative

Formulas

Quadratic Formula—

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Discriminant—

\Delta = b^2 - 4ac

Square Root Property—

x^2 = p \implies x = \pm\sqrt{p}

Completing the Square—

x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4}

Absolute Value Equation—

|p| = b \implies p = b \text{ or } p = -b

Absolute Value Inequalities—

|p| < b \implies -b < p < b

Vieta's Formulas (Quadratic)—

x_1 + x_2 = -\frac{b}{a}

Vieta's Formulas (General)—

r_1 + r_2 + \cdots + r_n = -\frac{a_{n-1}}{a_n}

See All Algebra Definitions →

See All Algebra Formulas →

What an Equation Is

Six lines of mathematics, all built from the same few symbols. Which of them are equations?

Some contain an unknown, some do not; some look like statements, some like formulas. On the page they are easy to confuse.

line	written
A	3 + 4 = 7
B	3 + 4 = 8
C	2x + 1
D	2x + 1 = 9
E	x + x = 2x
F	2x + 1 > 9

Four are equations — A, B, D and E — whether true, false or true only sometimes; C and F are not.

line	two sides joined by =?	what it claims	equation?
A: 3 + 4 = 7	yes	true, always	yes
B: 3 + 4 = 8	yes	false, always	yes
C: 2x + 1	no	nothing: it is a quantity	no
D: 2x + 1 = 9	yes	true only when x = 4	yes
E: x + x = 2x	yes	true for every x	yes
F: 2x + 1 > 9	no, joined by >	a comparison, not an equality	no

C has no equals sign, so it claims nothing; F has a sign, but the wrong one. A false line like B still counts.

An expression like

2x + 1

is not an equation. It has no equality sign and makes no claim. It can be evaluated, simplified, or factored, but it cannot be "solved" because there is no condition to satisfy. The distinction matters: solving requires an equation, not just an expression.

2x + 1 > 9

makes a claim too, but it compares sizes instead of asserting sameness — it is an inequality. And B is an equation even though it is false: being an equation is about the form of the claim, not about whether the claim holds.

The rule: two expressions joined by an equals sign, asserting that they have the same value.

Think of a balance scale: an equation says the two pans weigh the same. Whether the scale actually balances is a separate question.

An equation consists of two expressions joined by an equality sign. The expression to the left of

=

is the left-hand side, and the expression to the right is the right-hand side. Together they form a single assertion: whatever the left side evaluates to, the right side evaluates to the same thing.

The definition

An equation is a statement that two expressions are equal, written with the equality sign

=

between them.

2x + 1 = 9

is an equation;

2x + 1

on its own and

2x + 1 > 9

are not.

An equation is a claim: always true, always false, or true only for certain values of the unknown.

The statement

3 + 4 = 7

is an equation that is unconditionally true. The statement

3 + 4 = 8

is an equation that is unconditionally false. Neither of these involves an unknown — they are simply claims about arithmetic facts. The equations that drive algebra are different: they contain one or more unknowns (typically denoted

x

y

z

), and their truth depends on what values those unknowns take. The equation

2x + 1 = 9

is true when

x = 4

and false for every other value of

x

x = 3: true at one input, false everywhere else

The left side x is drawn as a curve and the right side 3 as a level; the equation asserts that they are equal, and that happens at exactly one input, where the curve crosses the level. At every other x the two sides differ, so the statement is false there. Move the marble along the axis and try other equations on the equation visual explorer.

Solving an equation means finding every input at which its assertion is true.

The rejected lines are not useless: each belongs to its own toolkit.

An expression like

2x + 1

is simplified, factored or evaluated, but never solved. An inequality like

2x + 1 > 9

is solved too, with an answer that is usually a whole range of values — the subject of the inequalities pages. For equations, the next section on solutions and solution sets says what "solved" means.

The rest of the page covers solutions and solution sets, types of equations by how many solutions they have, equivalent equations, and the main families of algebraic equations.

Solutions and Solution Sets

A solution to an equation is a value of the unknown that makes both sides equal. When

x = 4

is substituted into

2x + 1 = 9

, the left side becomes

2(4) + 1 = 9

, which matches the right side. The value

x = 4

satisfies the equation.

The solution set is the collection of all values that satisfy the equation. Some equations have a single solution, others have several, and some have infinitely many. The solution set of

x^2 = 4

\{-2, 2\}

because both values produce

4

when squared. The solution set of

x^2 = -1

over the real numbers is empty — no real number squared gives

-1

.

Set-builder notation provides a compact way to describe solution sets:

\{x \in \mathbb{R} : 2x + 1 = 9\}

reads "the set of all real numbers

x

such that

2x + 1 = 9

." For the equation above, this set contains only the element

4

x² + 4 = 0: the empty solution set

The curve x² + 4 never comes down to the level 0, so no real input makes the two sides equal: the solution set over ℝ is empty, just as for the section's x² = −1. Lower the curve until it touches or crosses the level, and the set gains one or two members, as x² = 4 has {−2, 2}. Try it on the equation visual explorer.

An empty solution set is still an answer: the equation has no solutions in that number system.

Types of Equations by Solution Behavior

Every equation in one variable falls into one of three categories based on how its solution set behaves.

A conditional equation is true for particular values and false for the rest. Most equations encountered in practice are conditional:

5x - 3 = 12

holds only when

x = 3

, and

x^2 - 5x + 6 = 0

holds only when

x = 2

x = 3

. The task of "solving" an equation almost always refers to finding the solutions of a conditional equation.

An identity is true for every permissible value of the variable. The equation

2(x + 1) = 2x + 2

holds regardless of what

x

is, because both sides are algebraically identical after expansion. Identities are not solved — they are verified. They express structural facts about expressions rather than constraints on unknowns.

A contradiction is true for no value of the variable. The equation

x + 1 = x + 3

simplifies to

1 = 3

, a false statement independent of

x

. No substitution can rescue it. The solution set is empty.

Type	True for	Example
Conditional	specific values only	5x − 3 = 12 → x = 3
Identity	all permissible values	2(x + 1) = 2x + 2
Contradiction	no value	x + 1 = x + 3

Equivalent Equations

Two equations are equivalent when they share the same solution set. The equations

2x = 10

and

x = 5

are equivalent because every value satisfying one satisfies the other. Solving an equation amounts to transforming it, step by step, into a simpler equivalent equation from which the solution is obvious.

RULE

Certain operations always produce equivalent equations. Adding or subtracting the same quantity on both sides preserves the solution set, as does multiplying or dividing both sides by any nonzero constant.

These reversible operations are the backbone of algebraic manipulation.

Pitfall: Dividing by the variable

Dividing

x^2 = 3x

x

gives

x = 3

and loses

x = 0

, where both sides equal

0

. Moving every term to one side,

x(x - 3) = 0

, gives

x = 0

x = 3

Other operations carry risk. Multiplying both sides by an expression containing the variable may introduce values that were not solutions of the original equation. Squaring both sides can do the same. The equation

x = 3

has one solution, but squaring gives

x^2 = 9

, whose solution set

\{-3, 3\}

is larger. The extra value

x = -3

is an extraneous solution — it satisfies the transformed equation but not the original. Any solving method that involves a non-reversible step demands verification: every candidate must be substituted back into the original equation to confirm it is genuine.

Operation	Reversible?	Risk
Add or subtract same quantity on both sides	✓	none
Multiply or divide both sides by nonzero constant	✓	none
Multiply both sides by an expression containing the variable	✗	may introduce extraneous solutions
Square both sides	✗	may introduce extraneous solutions

Reversible steps preserve the solution set; after any step that cannot be undone, check every candidate in the original equation.

Algebraic Equations

An algebraic equation is built entirely from variables, constants, and the operations of addition, subtraction, multiplication, division, and raising to integer powers. No transcendental functions — no sines, no logarithms, no exponentials with variable exponents — appear in a purely algebraic equation.

The degree of an algebraic equation is the highest power of the unknown that appears after the equation is cleared of fractions and simplified. Degree determines both the complexity of the equation and the maximum number of solutions it can have. A first-degree equation is called linear, a second-degree equation is quadratic, a third-degree equation is cubic, a fourth-degree equation is quartic, and the pattern continues with quintic, sextic, and so on.

Algebraic equations form the core of this section. Each degree class has its own structure, its own solving techniques, and in some cases its own dedicated formulas. The pages below treat the major classes individually, beginning with the simplest and progressing through increasingly complex types.

Degree	Name
1	Linear
2	Quadratic
3	Cubic
4	Quartic
5	Quintic
6	Sextic

Linear Equations

A linear equation in one variable takes the form

ax + b = 0

where

a \neq 0

. The variable appears only to the first power, and the equation has exactly one solution:

x = -\frac{b}{a}

. There are no special cases regarding the number of solutions — as long as

a

is nonzero, a unique answer exists.

Solving a linear equation requires nothing beyond the properties of equality: isolate the variable by undoing the operations applied to it, working in reverse order. When the equation involves parentheses, the distributive property clears them first. When fractions appear with numerical denominators, multiplying through by the least common denominator converts the equation to integer coefficients without changing its type. The mechanics are straightforward, but they establish the logical framework — preserving equivalence through reversible operations — that every other equation type builds upon.

Pitfall: Reading 0 = 0 as x = 0

Simplifying

2(x + 1) = 2x + 2

0 = 0

and writing

x = 0

reports a single solution. The statement

0 = 0

no longer contains

x

and is always true, so the solution set is all of

\mathbb{R}

The structure breaks only when the variable terms cancel entirely during simplification. If the remaining statement is true (like

0 = 0

), the original equation is an identity. If it is false (like

0 = 5

), the equation is a contradiction. These edge cases aside, first-degree equations are the most predictable objects in algebra.

3x − 2 = 4: one crossing

The line y = 3x − 2 meets the level y = 4 exactly once, at x = 2, and the marker sits on that crossing: a linear equation has one solution because a non-horizontal line crosses any level exactly once. Change the slope or the level and the single crossing simply moves. Drag the level line on the equation visual explorer.

One unknown, one power, one crossing: that is the whole theory of the linear case.

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Quadratic Equations

A quadratic equation has the form

ax^2 + bx + c = 0

with

a \neq 0

. The presence of

x^2

introduces a fundamental shift: the equation can have two solutions, one solution, or — over the real numbers — no solutions at all. The quantity

\Delta = b^2 - 4ac

, called the discriminant, settles the question without solving. When

\Delta

is positive, two distinct real roots exist. When

\Delta

equals zero, a single repeated root appears. When

\Delta

is negative, no real roots exist, though two complex conjugate roots do.

Three solving methods cover all cases. Factoring works when the quadratic decomposes into a product of two binomials with rational coefficients, but not every quadratic factors neatly. Completing the square works universally by rewriting the left side as a perfect square, and this technique is what produces the quadratic formula:

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Learn more about this formula: Quadratic Formula →

Browse all algebra formulas →

Pitfall: Dropping the sign of b

For

x^2 - 5x + 6 = 0

, writing

-b

-5

gives

x = \frac{-5 \pm 1}{2}

, that is

-2

and

-3

. Here

b = -5

, so

-b = 5

and

x = \frac{5 \pm 1}{2}

x = 3

x = 2

The formula accepts any coefficients and delivers the roots directly, making it the default tool when factoring is not immediate. Vieta's formulas offer a complementary perspective: the sum of the two roots equals

-\frac{b}{a}

and their product equals

\frac{c}{a}

, connecting solutions back to the coefficients that generated them.

To work through a specific equation, the quadratic equation solver computes the discriminant, applies the quadratic formula and graphs the parabola step by step.

Discriminant Δ = b² − 4ac	Real roots
Δ > 0	two distinct real roots
Δ = 0	one repeated real root
Δ < 0	no real roots; two complex conjugate roots

x² − 4 = 0: two crossings

The parabola meets the axis at x = −2 and x = 2, so the equation has two real solutions; raise the level above the vertex and the crossings vanish, set it on the vertex and they merge into one. The number of solutions is the number of times the curve meets the line, which is what the discriminant counts. Slide the level through the vertex on the equation visual explorer.

The discriminant is the algebraic name for how the level line sits relative to the vertex.

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Polynomial Equations

A polynomial equation of degree

n

has the general form

a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0

with

a_n \neq 0

. The Fundamental Theorem of Algebra guarantees exactly

n

roots in

\mathbb{C}

when counted with multiplicity, but finding those roots grows sharply harder as the degree increases.

For cubics and quartics, closed-form solutions exist — Cardano's formula for degree three and Ferrari's method for degree four — though both are cumbersome and rarely used in practice.

THEOREMAbel–Ruffini theorem

The Abel–Ruffini theorem draws a hard boundary at degree five: no general formula built from radicals and arithmetic can solve every quintic or higher-degree polynomial.

Specific equations beyond degree four may still yield to factoring, the rational root theorem, or substitution tricks, but there is no universal algebraic recipe.

The practical strategy for polynomial equations of any degree begins with the rational root theorem: test the finite list of candidate rational roots, and each confirmed root reduces the degree by one through polynomial division. Once the degree drops to two, the quadratic formula finishes the job. When no rational roots exist, numerical methods provide approximations to any desired accuracy.

For equations up to degree four, the polynomial equation solver runs this strategy step by step: rational root candidates, synthetic division, and a numerical fallback for irrational roots.

x³ − 3x = 0: three crossings

The cubic crosses the axis three times, at −√3, 0 and √3, the maximum a degree-three equation allows; shift the level up and one crossing at a time disappears until a single one remains, never none, because a cubic always crosses every level. Degree bounds the count, shape decides it. Move the level and count the marbles on the equation visual explorer.

The Fundamental Theorem counts complex roots; the picture shows how many of them are real.

The theorem closes one door at degree five. The card shows which door, and which stay open.

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Polynomial equations

The Abel–Ruffini Theorem

A hard boundary at degree five.

The theorem says

No general formula built from radicals and arithmetic can solve every polynomial equation of degree five or higher.

✕It does not say

Equations of degree five or higher cannot be solved.

Reading "no general formula" as "no solution"

Wrong workingThe equation is a quintic, so its roots cannot be found.

Right workingSpecific equations may still yield to factoring, the rational root theorem or substitution tricks; each confirmed root lowers the degree by one, and numerical methods approximate the rest to any accuracy.

Below degree five: closed forms exist, Cardano’s formula for cubics and Ferrari’s method for quartics, cumbersome and rarely used. The Fundamental Theorem of Algebra still guarantees exactly

n

complex roots.

Abel–Ruffini: what it does not say·/algebra/equationsLearn Math Class

No universal recipe, but still a strategy: rational root candidates, division, and a numerical fallback.

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Rational Equations

A rational equation contains at least one fraction whose denominator involves the variable. The equation

\frac{3}{x - 1} + \frac{2}{x + 4} = 1

is rational because

x

appears below the fraction bar, and the values

x = 1

and

x = -4

must be excluded before any solving begins — substituting either would produce division by zero.

The standard technique is to multiply both sides by the least common denominator of all fractions, clearing every denominator in one step. What remains is a polynomial equation, solvable by the methods of earlier sections. The catch is that this multiplication is not always reversible: if the LCD equals zero at some value of

x

, that value may emerge as a solution of the polynomial equation even though it was never in the domain of the original. These extraneous solutions must be filtered out by checking each candidate against the domain restrictions.

The risk of extraneous solutions is not a minor technicality — it is the defining feature of rational equations. Every solution procedure must end with verification. A candidate that zeros out any denominator in the original equation is rejected, regardless of how cleanly it emerged from the algebra.

Excluded values first, solutions second

The section's equation 3/(x − 1) + 2/(x + 4) = 1. Its left side is undefined at x = 1 and x = −4, where the graph breaks at the dashed lines; those inputs are ruled out before solving. Clearing denominators gives x² − 2x − 14 = 0, so x = 1 ± √15, and neither is an excluded value: both are the points where the curve meets y = 1.

Any candidate that lands on an excluded value is extraneous and must be discarded.

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Absolute Value Equations

An absolute value equation involves the absolute value function

|f(x)|

applied to an expression containing the unknown. The absolute value of a number is its distance from zero, so

|f(x)| = k

asks: where is

f(x)

exactly

k

units from zero? When

k > 0

, the answer splits into two cases —

f(x) = k

and

f(x) = -k

— each generating its own equation to solve. When

k = 0

, a single equation

f(x) = 0

remains. When

k < 0

, no solution exists because distance is never negative.

Pitfall: Forgetting the negative case

Solving

|x - 2| = 5

x - 2 = 5

gives only

x = 7

. The case

x - 2 = -5

adds

x = -3

, and

|-3 - 2| = 5

as well, so both

x = 7

and

x = -3

are solutions.

This case-splitting approach extends to more complex configurations. The equation

|f(x)| = |g(x)|

decomposes into

f(x) = g(x)

and

f(x) = -g(x)

. Equations with multiple absolute value terms require identifying the critical points where each expression inside

|\;|

changes sign, then solving on each interval separately using the piecewise definition. The geometric reading — absolute value as distance on the number line — often provides a faster route to the answer than purely mechanical case analysis.

k in \|f(x)\| = k	Solutions
k > 0	f(x) = k or f(x) = −k
k = 0	f(x) = 0
k < 0	no solution

|x| = 3: the V crosses the level twice

The V-shaped graph of |x| meets the level y = 3 at x = −3 and x = 3: the two cases of the case-splitting method are the two arms of the V. Lower the level to 0 and the crossings merge at the corner; lower it further and there are none. Watch the two solutions appear and disappear on the equation visual explorer.

Each arm of the V is one of the two linear equations the case split produces.

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Equations Beyond Algebra

Equations arise throughout mathematics, and many types fall outside the algebraic family because they involve operations or functions that algebra alone cannot handle. Each of these types has a dedicated treatment elsewhere on this site, and the links below point to those pages.

Exponential equations place the unknown in the exponent:

2^x = 16

3^{2x-1} = 5

. When the bases can be matched, the exponents are set equal directly. When they cannot, logarithms provide the bridge.

Logarithmic equations contain the unknown inside a logarithm:

\log_2(x + 3) = 5

. Converting to exponential form —

x + 3 = 2^5

— is the standard first move. Domain restrictions apply because logarithms are defined only for positive arguments.

Radical equations involve the unknown under a root sign:

\sqrt{x + 5} = x - 1

. Isolating the radical and raising both sides to the appropriate power removes the root but may introduce extraneous solutions, making verification mandatory.

Systems of linear equations involve multiple linear equations with multiple unknowns solved simultaneously. Gaussian elimination, matrix methods, and determinant-based formulas handle these systematically.

Trigonometric equations contain the unknown inside trigonometric functions like

\sin

\cos

, or

\tan

. The periodic nature of these functions typically produces infinitely many solutions.

Differential equations relate a function to its derivatives. They belong to calculus and arise whenever a quantity is defined by its rate of change rather than its value directly.

Type	Form	Key technique
Exponential	unknown in exponent (e.g. 2ˣ = 16)	match bases or apply logarithms
Logarithmic	unknown inside log (e.g. log₂(x+3) = 5)	convert to exponential form
Radical	unknown under root (e.g. √(x+5) = x−1)	isolate, raise to power, verify
Systems (linear)	multiple equations, multiple unknowns	Gaussian elimination, matrix methods
Trigonometric	unknown inside sin, cos, tan, …	periodicity → infinitely many solutions
Differential	function and its derivatives	calculus methods

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Before you read

Equations beyond the algebraic families

Once the unknown sits in an exponent, inside a logarithm, under a root or across several equations, new methods take over.

Equations beyond algebra6 kinds

Exponential

Unknownin the exponent

Methodmatch bases, otherwise take logs

Logarithmic

Unknowninside a log

Methodconvert to exponential form

argument must be positive

Radical

Unknownunder a root

Methodisolate, then raise to a power

check for extraneous roots

Linear systems

Unknownsseveral at once

MethodGaussian elimination or matrices

Trigonometric

Featureperiodic: infinitely many solutions

Differential

Relatesa function to its derivatives

Takeaway

Same goal, new tools

Each kind still asks for every value that makes the statement true; the tools change with where the unknown hides.

Equations beyond algebra·/algebra/equationsLearn Math Class

Summary of Algebraic Equation Types

The table below collects the five algebraic equation classes covered on this page. For each class, it shows the canonical form, the maximum number of real solutions, and the primary technique used to solve it. Use it as a reference card.

Class	General form	Max real solutions	Primary technique
Linear	ax + b = 0	1	isolate the variable
Quadratic	ax² + bx + c = 0	2	quadratic formula / discriminant
Polynomial (degree n)	aₙxⁿ + ⋯ + a₀ = 0	n	rational roots + division
Rational	fractions with variable denominators	degree-dependent	clear denominators, verify
Absolute value	\|f(x)\| = k	case-dependent	split into cases

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Before you read

Five algebraic equation families

Each family below has its own form, its own count of solutions, and its own method. Two of them come with a check that cannot be skipped. Open an item for the longer explanation.

Algebraic equations5 families

Linear

ax + b = 0

Solutionsexactly one:

x = -b/a

Methodisolate the variable

Undo the operations applied to the variable in reverse order. Parentheses are cleared with the distributive property; numerical denominators are cleared by multiplying through by the least common denominator.

Edge casevariable cancels

If the remaining statement is true, like

0 = 0

, the equation is an identity. If it is false, like

0 = 5

, it is a contradiction.

Quadratic

ax^2 + bx + c = 0

Solutions2, 1 or none over

\mathbb{R}

The discriminant

\Delta = b^2 - 4ac

decides: two real roots when

\Delta > 0

, one repeated root when

\Delta = 0

, none when

\Delta < 0

(two complex conjugate roots instead).

Methodfactor, complete the square, or the formula

Factoring works when the quadratic splits into binomials with rational coefficients. Completing the square always works and produces the quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Polynomial

a_n x^n + \cdots + a_0 = 0

Solutionsexactly

n

\mathbb{C}

, with multiplicity

Methodrational roots, then divide

Test the candidate rational roots; each confirmed root lowers the degree by one. At degree two the quadratic formula finishes. With no rational roots, numerical methods approximate.

Limitno general formula from degree 5

Cardano (degree 3) and Ferrari (degree 4) give closed forms. The Abel–Ruffini theorem rules out a general radical formula for degree five and higher.

Rational

P(x)/Q(x)

Methodmultiply by the LCD

Clearing every denominator in one step leaves a polynomial equation, solvable by the methods of the earlier families.

Thencheck each candidate

Multiplying by the LCD is not always reversible, so it can create extraneous solutions. A candidate that makes any original denominator zero is rejected.

Exclude every zero of a denominator

Absolute value

|f(x)| = k

k > 0

f(x) = k

f(x) = -k

k = 0

f(x) = 0

More termssplit at critical points

With several absolute values, find where each inside expression changes sign and solve on each interval with the piecewise definition.

|f(x)| = |g(x)|

splits into

f(x) = g(x)

and

f(x) = -g(x)

No solution when $k < 0$

equation familya restriction applies

Takeaway

Verify where the method can mislead

Linear and quadratic equations need no checking beyond arithmetic. Rational equations must be checked against their excluded values, and absolute value equations split into cases that each need solving.

Five families of algebraic equations·/algebra/equationsLearn Math Class

Common Pitfalls

Five mistakes cause most wrong answers in the equation types on this page. Four of them are marked where they happen in the sections above; here each is worked through in full, with the wrong and the right working side by side.

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1Dividing by the variable

Both sides of an equation share a factor that contains the unknown, and the factor is cancelled by dividing both sides by it.

Start

x^2 = 3x

Wrong turn

Error here

\implies x = 3

Right path

Move every term to one side and factor.

x^2 - 3x = 0

\implies x(x - 3) = 0

\implies x = 0 \text{ or } x = 3

Why it happensDividing by a constant is always safe, and dividing by

x

looks like the same step. But when

x = 0

it is a division by zero, so the solution

x = 0

is thrown away without notice.

How to checkSubstitute

x = 0

into the original equation. Both sides give

0

, so it is a solution the wrong working lost.

2Skipping the check after squaring

A radical equation is squared to remove the root, and every solution of the squared equation is reported as an answer.

Start

\sqrt{x + 5} = x - 1

Wrong turn

Error here

\implies x + 5 = x^2 - 2x + 1

\implies x^2 - 3x - 4 = 0

\implies x = 4 \text{ or } x = -1

Right path

Squaring is not reversible, so each candidate goes back into the original equation. For

x = 4

\sqrt{9} = 3

and

4 - 1 = 3

, which match. For

x = -1

\sqrt{4} = 2

but

-1 - 1 = -2

, which do not. The only solution is

x = 4

Why it happensThe squared equation is an ordinary quadratic, and its two clean roots look final. Squaring also accepts values where the two sides are opposites, since

2^2 = (-2)^2

How to checkSubstitute every candidate into the original equation, not into the squared one.

3Reading 0 = 0 as x = 0

The variable terms cancel while simplifying, the statement

0 = 0

is left, and it is read as the answer

x = 0

Start

2(x + 1) = 2x + 2

Wrong turn

Error here

\implies 2x + 2 = 2x + 2

\implies 0 = 0

\implies x = 0

Right path

0 = 0

contains no

x

and is true, so the original equation holds for every value of

x

. It is an identity, and the solution set is

\mathbb{R}

Why it happensMost solving ends with a line of the form

x = \ldots

, and a lone

0

on the last line looks like the value of

x

How to checkSubstitute a value other than

0

. For

x = 5

, both sides give

12

, so

x = 0

is not the only solution.

4Dropping the sign of b

The quadratic formula is used with a negative coefficient

b

, and

-b

is written with the wrong sign.

Wrong turn

Error here

For

x^2 - 5x + 6 = 0

, with

a = 1

b = -5

c = 6

and

\Delta = 25 - 24 = 1

x = \frac{-5 \pm 1}{2}

\implies x = -2 \text{ or } x = -3

Right path

-b = -(-5) = 5

, so

x = \frac{5 \pm 1}{2}

\implies x = 3 \text{ or } x = 2

Why it happensThe minus sign in front of

b

in the formula and the minus sign inside

b

itself are easy to merge into one, especially when

b

is copied straight from the equation.

How to checkSubstitute a root.

x = -2

gives

4 + 10 + 6 = 20

, not

0

;

x = 2

gives

4 - 10 + 6 = 0

. By Vieta's formulas the roots must also add up to

-\frac{b}{a} = 5

5Forgetting the negative case

An absolute value equation

|f(x)| = k

with

k > 0

is solved by dropping the bars, so only the case

f(x) = k

is solved.

Start

|x - 2| = 5

Wrong turn

Error here

\implies x - 2 = 5

\implies x = 7

Right path

Both cases are needed.

x - 2 = 5 \implies x = 7

x - 2 = -5 \implies x = -3

Why it happensThe bars look like a pair of brackets, and removing them seems harmless. But a number and its opposite have the same absolute value, so each value of

k

comes from two values of

f(x)

How to checkUse the distance reading.

|x - 2| = 5

asks for the points at distance

5

from

2

on the number line, and there are two of them,

7

and

-3

Common pitfalls in solving equations·/algebra/equationsLearn Math Class

Two of the slips lose solutions: dividing by

x

drops

x = 0

, and taking only the positive case of

|x - 2| = 5

drops

x = -3

. One keeps a false solution, because squaring can add a root that the original equation rejects. The other two misread a line of working:

0 = 0

means that every

x

is a solution, and

-b

for

b = -5

5

. Substituting the candidates, and the values the working left out, back into the original equation catches every one of them.

Equations FAQ

Is a false statement like $3 + 4 = 8$ still an equation?

Yes. Being an equation is a matter of form: two expressions joined by

=

. The line

3 + 4 = 8

has that form, so it is an equation, just a false one. An equation can be always true, like

3 + 4 = 7

, always false, like

3 + 4 = 8

, or true only for some values of the unknown, like

2x + 1 = 9

x = 4

. By contrast,

2x + 1 > 9

is not an equation: it compares sizes with

>

, so it is an inequality.Read more →

Does $x^2 = -1$ have a solution?

Not over the real numbers. No real number squared gives

-1

, so the solution set over

\mathbb{R}

is empty, and an empty solution set is a complete answer, not a failure to solve. The answer depends on the number system: over the complex numbers,

x^2 = -1

has the two solutions

i

and

-i

. This is why a quadratic with a negative discriminant has no real roots but two complex conjugate roots.Read more →

Which equations need their solutions checked?

Any equation solved with a step that cannot be undone. Adding or subtracting the same quantity, and multiplying or dividing by a nonzero constant, keep the solution set. Multiplying by an expression that contains the variable, as when clearing the denominators of a rational equation, and squaring both sides, as in a radical equation, can add extraneous solutions, so every candidate goes back into the original equation. Dividing by the variable does the opposite:

x^2 = 3x

divided by

x

loses

x = 0

.Read more →

Is $2^x = 16$ an algebraic equation?

No. An algebraic equation uses only addition, subtraction, multiplication, division and integer powers of the unknown, so the unknown can be a base but never an exponent. In

2^x = 16

the unknown is the exponent, which makes it an exponential equation, solved by matching bases (

16 = 2^4

, so

x = 4

) or with logarithms. Logarithmic and trigonometric equations fall outside the algebraic family for the same reason: the unknown sits inside a function that algebra alone cannot undo.Read more →

How many solutions can a rational or absolute value equation have?

It depends on what the equation contains, which is why the summary table gives no fixed maximum. Clearing the denominators of a rational equation leaves a polynomial equation, so unless the variable cancels out entirely, that polynomial's degree caps the count, minus any candidate that is an excluded value:

\frac{3}{x - 1} + \frac{2}{x + 4} = 1

becomes

x^2 - 2x - 14 = 0

and keeps both roots

1 \pm \sqrt{15}

. An equation

|f(x)| = k

with

k > 0

splits into

f(x) = k

and

f(x) = -k

, so it has up to two solutions when

f

is linear and up to four when

f

is quadratic.Read more →

Equations

The Language of Mathematical Conditions

Key Terms

Equations and Expressions

Solutions

Equation Classification

Supporting Concepts

Formulas

What an Equation Is

Solutions and Solution Sets

Types of Equations by Solution Behavior

Equivalent Equations

Algebraic Equations

Linear Equations

Quadratic Equations

Polynomial Equations

The Abel–Ruffini Theorem

Rational Equations

Absolute Value Equations

Equations Beyond Algebra

Summary of Algebraic Equation Types

Common Pitfalls

Equations FAQ

Is a false statement like 3+4=83 + 4 = 83+4=8 still an equation?

Does x2=−1x^2 = -1x2=−1 have a solution?

Which equations need their solutions checked?

Is 2x=162^x = 162x=16 an algebraic equation?

How many solutions can a rational or absolute value equation have?

Is a false statement like $3 + 4 = 8$ still an equation?

Does $x^2 = -1$ have a solution?

Is $2^x = 16$ an algebraic equation?