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Algebra Formulas

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Logarithm Rules
Identities & Factoring
Exponent Rules
Radical Rules
Polynomial Theorems
48 formulas

Equations

(6 formulas)

Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
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Gives the two solutions of any quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 directly from its coefficients. The ±\pm produces two values: one using ++, the other using −-. The expression under the square root, b2−4acb^2 - 4ac, is the discriminant — it determines whether the roots are real and distinct, real and equal, or complex.
Function machine: a, b, c → √Δ → x₁, x₂a, b, c√Δx₁, x₂compute discriminant b² − 4acapply ± and divide by 2aQuadratic Formula
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Discriminant

Δ=b2−4ac\Delta = b^2 - 4ac
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The discriminant of a quadratic ax2+bx+c=0ax^2 + bx + c = 0 determines the nature of its roots. If Δ>0\Delta > 0: two distinct real roots. If Δ=0\Delta = 0: one repeated real root. If Δ<0\Delta < 0: two complex conjugate roots.
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Square Root Property

x2=p  ⟹  x=±px^2 = p \implies x = \pm\sqrt{p}
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If a squared expression equals a constant, the variable equals the positive or negative square root of that constant. This is the simplest method for solving quadratics with no linear term.
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Completing the Square

x2+bx=(x+b2)2−b24x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4}
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Rewrites a quadratic expression as a perfect square minus a constant. Take half the coefficient of xx, square it, add and subtract.
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Absolute Value Equation

∣p∣=b  ⟹  p=b or p=−b(b>0)|p| = b \implies p = b \text{ or } p = -b \quad (b > 0)
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An absolute value equation splits into two cases: the expression inside equals the positive value, or the expression inside equals the negative value.
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Absolute Value Inequalities

∣p∣<b  ⟹  −b<p<b∣p∣>b  ⟹  p<−b or p>b|p| < b \implies -b < p < b \qquad |p| > b \implies p < -b \text{ or } p > b
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A less-than inequality produces a compound inequality (a bounded interval). A greater-than inequality produces a disjunction (two unbounded rays).
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Logarithm Rules

(8 formulas)

Product Rule (Logarithms)

log⁡a(xy)=log⁡a(x)+log⁡a(y)\log_a(xy) = \log_a(x) + \log_a(y)
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The logarithm of a product equals the sum of the logarithms. This converts multiplication inside the argument into addition outside.
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Quotient Rule (Logarithms)

log⁡a(xy)=log⁡a(x)−log⁡a(y)\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)
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The logarithm of a quotient equals the difference of the logarithms. Division inside the argument becomes subtraction outside.
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Power Rule (Logarithms)

log⁡a(xn)=n⋅log⁡a(x)\log_a(x^n) = n \cdot \log_a(x)
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An exponent inside the argument moves out front as a multiplier. This is the key property that makes logarithms useful for simplifying expressions with exponents.
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Change of Base Formula

log⁡a(x)=log⁡b(x)log⁡b(a)\log_a(x) = \frac{\log_b(x)}{\log_b(a)}
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Converts a logarithm from one base to another. Most commonly used with b=10b = 10 or b=eb = e to evaluate logarithms on a calculator, which only has log⁡\log and ln⁡\ln keys.
Function machine: log_a(x) → log_b(x) and log_b(a) → log_b(x) / log_b(a)log_a(x)log_b(x) and log_b(a)log_b(x) / log_b(a)compute log_b of both x and adivideChange of Base
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Logarithm of the Base

log⁡a(a)=1\log_a(a) = 1
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The logarithm of the base itself always equals 1, because a1=aa^1 = a.
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Logarithm of One

log⁡a(1)=0\log_a(1) = 0
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The logarithm of 1 is always 0, regardless of the base, because a0=1a^0 = 1.
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Logarithm of an Exponential

log⁡a(ax)=x\log_a(a^x) = x
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Applying a logarithm to its own base's exponential cancels both operations, returning the exponent. The log "undoes" the exponential.
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Exponential of a Logarithm

alog⁡a(x)=xa^{\log_a(x)} = x
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Applying an exponential to its own base's logarithm cancels both operations, returning the argument. The exponential "undoes" the log.
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Identities & Factoring

(11 formulas)

Difference of Squares

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)
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A difference of two perfect squares factors into the product of a sum and a difference. This is the most frequently used factoring identity in algebra.
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Square of a Sum

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
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Squaring a binomial sum produces a trinomial: the square of each term plus twice their product.
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Square of a Difference

(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2
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Squaring a binomial difference produces a trinomial: the square of each term minus twice their product. The result is always positive — a squared quantity.
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Cube of a Sum

(a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
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Cubing a binomial sum. The coefficients follow the third row of Pascal's triangle: 1, 3, 3, 1.
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Cube of a Difference

(a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
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Cubing a binomial difference. Same coefficients as the cube of a sum, with alternating signs.
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Sum of Cubes

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)
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A sum of two cubes factors into a binomial times a trinomial. The trinomial factor a2−ab+b2a^2 - ab + b^2 is irreducible over the reals.
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Difference of Cubes

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
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A difference of two cubes factors into a binomial times a trinomial. Compare with the sum of cubes — the signs alternate in a predictable pattern: same, opposite, always positive.
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Trinomial Factoring Pattern

x2+(a+b)x+ab=(x+a)(x+b)x^2 + (a + b)x + ab = (x + a)(x + b)
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A monic quadratic trinomial factors into two binomials whose constants multiply to the constant term and add to the linear coefficient. This reverses the FOIL multiplication pattern.
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General Difference of Even Powers

x2n−a2n=(xn−an)(xn+an)x^{2n} - a^{2n} = (x^n - a^n)(x^n + a^n)
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Any difference of even powers splits as a difference of squares applied to the nn-th powers. Each factor may be factorable further depending on whether nn is even or odd.
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General Difference of Powers (odd n)

xn−an=(x−a)(xn−1+axn−2+⋯+an−1)x^n - a^n = (x - a)(x^{n-1} + ax^{n-2} + \cdots + a^{n-1})
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For any positive integer nn, the difference xn−anx^n - a^n has (x−a)(x - a) as a factor. The second factor is the sum of all terms xn−1−kakx^{n-1-k}a^k for k=0k = 0 to n−1n - 1.
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General Sum of Powers (odd n)

xn+an=(x+a)(xn−1−axn−2+⋯+an−1)x^n + a^n = (x + a)(x^{n-1} - ax^{n-2} + \cdots + a^{n-1})
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For odd nn, the sum xn+anx^n + a^n has (x+a)(x + a) as a factor. The second factor has alternating signs. This identity does not hold for even nn — a sum of even powers does not factor over the reals.
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Exponent Rules

(8 formulas)

Product Rule (Exponents)

am⋅an=am+na^m \cdot a^n = a^{m+n}
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Multiplying powers with the same base: keep the base, add the exponents.
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Quotient Rule (Exponents)

aman=am−n\frac{a^m}{a^n} = a^{m-n}
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Dividing powers with the same base: keep the base, subtract the exponents.
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Power of a Power

(am)n=amn(a^m)^n = a^{mn}
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Raising a power to another power: keep the base, multiply the exponents.
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Power of a Product

(ab)n=an⋅bn(ab)^n = a^n \cdot b^n
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A power applied to a product distributes to each factor.
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Power of a Quotient

(ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
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A power applied to a fraction distributes to numerator and denominator separately.
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Zero Exponent

a0=1(a≠0)a^0 = 1 \quad (a \neq 0)
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Any nonzero base raised to the zero power equals 1. This follows from the quotient rule: an/an=an−n=a0=1a^n / a^n = a^{n-n} = a^0 = 1.
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Negative Exponent

a−n=1an(a≠0)a^{-n} = \frac{1}{a^n} \quad (a \neq 0)
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A negative exponent means the reciprocal of the positive power. It does not make the result negative.
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Negative Exponent Flip

(ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n
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A negative exponent on a fraction inverts the fraction and makes the exponent positive.
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Radical Rules

(7 formulas)

Radical to Exponent Conversion

an=a1/n\sqrt[n]{a} = a^{1/n}
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The nn-th root of aa equals aa raised to the power 1/n1/n. This bridges radical notation and exponential notation, allowing all radical operations to be performed using exponent rules.
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Product Rule (Radicals)

abn=an⋅bn\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}
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The nn-th root of a product equals the product of the nn-th roots. Used to simplify radicals by factoring the radicand.
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Quotient Rule (Radicals)

abn=anbn\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}
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The nn-th root of a quotient equals the quotient of the nn-th roots.
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Power Rule (Radicals)

amn=am/n\sqrt[n]{a^m} = a^{m/n}
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Combines the radical to exponent conversion with the power of a power rule. The nn-th root of ama^m equals aa raised to the fraction m/nm/n.
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Nested Radicals

anm=amn\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}
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A radical inside a radical simplifies by multiplying the indices. In exponent form: (a1/n)1/m=a1/(mn)(a^{1/n})^{1/m} = a^{1/(mn)}.
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Even Root Identity

ann=∣a∣(n even)\sqrt[n]{a^n} = |a| \quad (n \text{ even})
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When the index is even, the nn-th root of ana^n returns the absolute value of aa, not aa itself. This is because even powers erase the sign: (−3)2=9(-3)^2 = 9, and 9=3\sqrt{9} = 3, not −3-3.
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Odd Root Identity

ann=a(n odd)\sqrt[n]{a^n} = a \quad (n \text{ odd})
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When the index is odd, the nn-th root of ana^n returns aa directly — no absolute value needed. Odd roots preserve sign.
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Polynomial Theorems

(8 formulas)

Remainder Theorem

P(x)=(x−c)⋅Q(x)+P(c)P(x) = (x - c) \cdot Q(x) + P(c)
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When a polynomial P(x)P(x) is divided by (x−c)(x - c), the remainder equals P(c)P(c) — the value of the polynomial evaluated at cc. No long division needed to find the remainder.
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Factor Theorem

(x−c) is a factor of P(x)  ⟺  P(c)=0(x - c) \text{ is a factor of } P(x) \iff P(c) = 0
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(x−c)(x - c) divides P(x)P(x) evenly if and only if cc is a root of PP. This is the remainder theorem with remainder equal to zero.
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Rational Root Theorem

If pq is a root: p∣a0 and q∣an\text{If } \frac{p}{q} \text{ is a root: } p \mid a_0 \text{ and } q \mid a_n
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For a polynomial with integer coefficients, any rational root p/qp/q (in lowest terms) must have pp dividing the constant term a0a_0 and qq dividing the leading coefficient ana_n. This limits the search for rational roots to a finite list of candidates.
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Vieta's Formulas (Quadratic)

x1+x2=−bax1⋅x2=cax_1 + x_2 = -\frac{b}{a} \qquad x_1 \cdot x_2 = \frac{c}{a}
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Relates the roots of a quadratic ax2+bx+c=0ax^2 + bx + c = 0 to its coefficients without solving the equation. The sum of the roots equals −b/a-b/a and the product equals c/ac/a.
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Binomial Theorem

(x+y)n=∑k=0n(nk)xn−kyk(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
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Expands any positive integer power of a binomial as a sum of n+1n + 1 terms. Each term is weighted by a binomial coefficient (nk)\binom{n}{k}, with the powers of xx decreasing and the powers of yy increasing.
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Binomial Coefficient

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}
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Counts the number of ways to choose kk items from nn items, and gives the coefficient of the kk-th term in the binomial expansion. Read "nn choose kk."
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Pascal's Rule

(nk)+(nk+1)=(n+1k+1)\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}
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Each entry in Pascal's triangle equals the sum of the two entries directly above it. This recurrence builds binomial coefficients row by row without computing factorials.
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Vieta's Formulas (General)

r1+r2+r3=−an−1r1r2r3=(−1)na0r_1 + r_2 + r_3 = -a_{n-1} \qquad r_1 r_2 r_3 = (-1)^n a_0
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Generalizes Vieta's formulas to polynomials of any degree (shown here for degree 3 with leading coefficient 1). The sum of the roots equals the negative of the second coefficient. The product of the roots equals the constant term times (−1)n(-1)^n.
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Equations
Quadratic FormulaDiscriminantSquare Root PropertyCompleting the SquareAbsolute Value EquationAbsolute Value Inequalities
Logarithm Rules
Product Rule (Logarithms)Quotient Rule (Logarithms)Power Rule (Logarithms)Change of Base FormulaLogarithm of the BaseLogarithm of OneLogarithm of an ExponentialExponential of a Logarithm
Identities & Factoring
Difference of SquaresSquare of a SumSquare of a DifferenceCube of a SumCube of a DifferenceSum of CubesDifference of CubesTrinomial Factoring PatternGeneral Difference of Even PowersGeneral Difference of Powers (odd n)General Sum of Powers (odd n)
Exponent Rules
Product Rule (Exponents)Quotient Rule (Exponents)Power of a PowerPower of a ProductPower of a QuotientZero ExponentNegative ExponentNegative Exponent Flip
Radical Rules
Radical to Exponent ConversionProduct Rule (Radicals)Quotient Rule (Radicals)Power Rule (Radicals)Nested RadicalsEven Root IdentityOdd Root Identity
Polynomial Theorems
Remainder TheoremFactor TheoremRational Root TheoremVieta's Formulas (Quadratic)Binomial TheoremBinomial CoefficientPascal's RuleVieta's Formulas (General)