Algebra Equation Solvers | Learn Math Class

Linear

Linear Equation Solver

ax + b = cx + d

A linear equation is one where the variable

x

appears only to the first power — no

x^2

, no

\sqrt{x}

, no

x

in a denominator. The solver expands brackets, distributes through factored expressions like

2(x + 3)

, and combines like terms on each side. It then moves all

x

-terms to the left and all constants to the right, and finally divides by the coefficient of

x

to isolate it. Equations that simplify to a true statement like

0 = 0

are flagged as identities (every

x

is a solution); equations that simplify to a false statement like

5 = 7

are flagged as contradictions (no solution). Final answers appear as exact reduced fractions whenever the division does not produce an integer, never as rounded decimals.

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Quadratic

Quadratic Equation Solver

ax^2 + bx + c = 0

A quadratic equation is one where the highest power of the variable is

2

. The solver rewrites the equation in standard form, then computes the discriminant

\Delta = b^2 - 4ac

and uses its sign to route the solution. A positive

\Delta

produces two distinct real roots via the quadratic formula

x = \dfrac{-b \pm \sqrt{\Delta}}{2a}

, with factoring shown alongside whenever

\Delta

is a perfect square. A zero

\Delta

produces a single repeated root and the corresponding perfect-square trinomial factorization. A negative

\Delta

means no real solutions exist, in which case the parabola is graphed showing the curve sitting entirely above or below the

x

-axis. Every solved equation also reports the parabola's vertex

(h, k)

and is accompanied by a full graph with axis-crossings, vertex, and equation label marked.

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Polynomial

Polynomial Equation Solver

a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0

A polynomial equation is the general form

a_n x^n + \dots + a_1 x + a_0 = 0

, where

n

is any positive integer — linear and quadratic equations are the special cases

n = 1

and

n = 2

. This solver handles degrees up to

4

. Linear and quadratic cases reuse their dedicated logic. For degrees

3

and

4

, the solver applies the rational root theorem to enumerate every candidate

\dfrac{p}{q}

(where

p

divides the constant term

a_0

and

q

divides the leading coefficient

a_n

), tests each by substitution, and uses synthetic division to factor each successful root out — leaving a polynomial of one degree less to recurse on. When no rational root exists, a Newton's-method fallback locates real irrational roots numerically. Special cases — equations that factor out a power of

x

, identities, contradictions — are recognized and reported separately.

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Rational

Rational Equation Solver

\dfrac{P(x)}{Q(x)} + \dfrac{R(x)}{S(x)} = \dots

A rational equation is one where the variable appears in a denominator — fractions whose tops or bottoms are polynomials in

x

, such as

\dfrac{x+1}{x-2} = 3

. The solver first identifies every value of

x

that makes any denominator zero; these are the restricted values that no solution may equal. It then computes the least common denominator across every fraction in the equation, multiplies both sides through by the LCD to clear all fractions, and solves the resulting polynomial equation (degrees

1

–

4

). Every candidate solution is then checked against the restricted set: any that would make a denominator zero is reported as extraneous and excluded from the final answer. Identities, contradictions, and the case where every candidate turns out to be extraneous are each labeled as their own step. Fractions are rendered in proper stacked form, not inline, so the structure stays readable while building.

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Radical

Radical Equation Solver

\sqrt[n]{P(x)} + \dots = \dots

A radical equation is one containing roots — square roots, cube roots, or higher-index roots — with the variable underneath, such as

\sqrt{x + 3} = 5

\sqrt[3]{2x - 1} = 4

. The solver isolates the radical term on one side, then raises both sides to the appropriate power (

2

for

\sqrt{\,}

3

for

\sqrt[3]{\,}

n

for

\sqrt[n]{\,}

) to eliminate it. The resulting linear or quadratic equation is solved with the corresponding routine. Because raising both sides to an even power can introduce false solutions, every candidate is substituted back into the original equation and verified; any that fail are reported as extraneous and discarded. Indices

2

3

, and arbitrary

n

are all supported, including equations where the radical appears with a coefficient or inside a more complex expression.

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Exponential

Exponential Equation Solver

a \cdot b^{f(x)} + c = d

An exponential equation is one where the variable appears in an *exponent* rather than as a base — the unknown is the power something is raised to, such as

2^x = 32

5 \cdot 3^{x+1} = 45

. The solver isolates the exponential term first by performing whatever additions, subtractions, and divisions are needed to leave a single

b^{(\dots)}

on one side. It then attempts base-matching: if the right-hand side can be rewritten as a power of the same base

b

(recognizing

8 = 2^3

when working with

2^x

), the exponents are set equal directly, yielding an exact integer or rational solution. When base-matching is not possible, the solver applies the natural logarithm to both sides, uses the power rule

\ln(b^x) = x \ln(b)

to bring the exponent down, and reduces the problem to a linear equation in

x

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Logarithmic

Logarithmic Equation Solver

\log_b(P(x)) = c

A logarithmic equation is one containing logarithms — the inverse of exponentials — with the variable inside the log argument, such as

\log_2(x + 1) = 5

\ln(3x) = 2

. Two cases are recognized. When both sides are logs of the same base,

\log_b(A) = \log_b(B)

, the equal-logarithms property applies: the arguments must be equal, so the equation reduces directly to

A = B

and is solved as a standard algebraic equation. When only one side is a log and the other is a number or expression, the solver isolates the log first, then converts to exponential form using the inverse relationship

\log_b(A) = c \iff A = b^c

, and solves the resulting equation. Domain restrictions on the original log arguments are kept in mind throughout — logarithms are only defined for positive inputs.

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Absolute Value

Absolute Value Equation Solver

|ax + b| = c

An absolute value equation is one containing the

|\cdot|

bars around an expression. The absolute value of a number is its distance from zero on the number line, so

|{-5}| = 5

and

|5| = 5

— the bars strip away the sign. The solver isolates the absolute value term first, then handles the special cases up front: if the right side equals

0

there is one solution where the inside equals

0

; if the right side is negative there is no solution at all (since

|\cdot| \geq 0

always). For the general case

|A| = c

with

c > 0

, the equation splits into two sub-equations —

A = c

(interior positive) and

A = -c

(interior negative) — each solved independently. The variant

|A| = |B|

where both sides are absolute values is also supported, with

A = B

and

A = -B

branches.

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Literal

Literal Equation Solver

F(a,\ b,\ c,\ \dots) = 0

A literal equation is a formula with several letters in it where you solve for *one* of the letters in terms of the others — like rearranging

A = \pi r^2

to get

r = \sqrt{A / \pi}

, or solving

F = ma

for

a = F/m

. These are the equations that appear in physics, geometry, and finance, where multiple quantities relate to each other by a fixed formula. The solver auto-detects every variable in the expression and presents them as buttons, letting you pick which one to solve for. The equation is then decomposed symbolically into the form

(\text{coefficient}) \cdot (\text{target}) + (\text{constant})

, where the coefficient and constant are themselves symbolic expressions in the remaining letters. The two sides are manipulated to isolate the target, returning a closed-form symbolic answer rather than a number.

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