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Algebra Formulas & Calculators

Understand algebra formulas with explanations, examples and interactive tools.

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Square of a Sum

\((a+b)^2=a^2+2ab+b^2\)

Expand the square of a binomial.

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Square of a Difference

\((a-b)^2=a^2-2ab+b^2\)

Expand the square of a difference.

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Difference of Squares

\(a^2-b^2=(a-b)(a+b)\)

Factor a difference of two squares.

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Product Identity

\((x+a)(x+b)=x^2+(a+b)x+ab\)

Expand two linear factors.

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Linear Equation

ax+b=c\Rightarrow x=\frac{c-b}{a}

Solve an equation in one variable when a ≠ 0.

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Exponent Product

a^m a^n=a^{m+n}

Same base: add exponents.

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Exponent Quotient

\frac{a^m}{a^n}=a^{m-n}

Same nonzero base: subtract exponents.

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Power of a Power

(a^m)^n=a^{mn}

Multiply exponents.

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Quadratic Discriminant

\(D=b^2-4ac\)

Determines nature of quadratic roots.

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Quadratic Roots

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

Find roots of ax²+bx+c=0, a ≠ 0.

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Sum of Quadratic Roots

\alpha+\beta=-\frac ba

Sum of roots of ax²+bx+c=0.

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Product of Quadratic Roots

\alpha\beta=\frac ca

Product of roots of ax²+bx+c=0.

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Arithmetic Progression nth Term

\(a_n=a+(n-1)d\)

Find the nth term of an arithmetic progression.

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Arithmetic Progression Sum

S_n=\frac n2[2a+(n-1)d]

Sum of first n terms of an arithmetic progression.

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Geometric Progression nth Term

a_n=ar^{n-1}

Find nth term of geometric sequence.

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Geometric Progression Sum

S_n=a\frac{r^n-1}{r-1}

Sum of n geometric terms when r ≠ 1; for r=1, S=na.

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Permutations

{}^nP_r=\frac{n!}{(n-r)!}

Ordered selections of r from n distinct objects.

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Combinations

{}^nC_r=\frac{n!}{r!(n-r)!}

Unordered selections of r from n distinct objects.

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Binomial General Term

T_{r+1}={}^{n}C_r a^{n-r}b^r

General term in expansion of (a+b)^n.

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