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Algebra Formulas Cheat Sheet (with Examples)

Ilichapishwa 2 Novemba 2026 · Na Sumzo team

An algebra formulas cheat sheet: lines, quadratics, factoring identities, exponent and log laws, sequences and interest — each with a quick example.

This algebra formulas cheat sheet collects the formulas used most in school algebra, each with a one-line example. Use it to revise — and to check which ones you can already rebuild from understanding rather than memory.

Lines

• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$. Through $(1, 3)$ and $(3, 7)$: $m = \frac{4}{2} = 2$.
• Slope-intercept form: $y = mx + b$, where $b$ is the y-intercept.
• Point-slope form: $y - y_1 = m(x - x_1)$.
• Parallel lines have equal slopes; perpendicular slopes multiply to $-1$.

Quadratics

• Standard form: $ax^2 + bx + c = 0$.
• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Discriminant: $\Delta = b^2 - 4ac$ — positive: two real roots; zero: one; negative: none that are real.
• Vertex: at $x = -\frac{b}{2a}$. For $y = x^2 - 4x + 3$, the vertex is at $x = 2$, $y = -1$.
• Sum and product of roots: $x_1 + x_2 = -\frac{b}{a}$, $x_1 x_2 = \frac{c}{a}$.

Factoring and expanding identities

• $(a + b)^2 = a^2 + 2ab + b^2$
• $(a - b)^2 = a^2 - 2ab + b^2$
• $a^2 - b^2 = (a + b)(a - b)$. Example: $x^2 - 9 = (x + 3)(x - 3)$.
• $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
• $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Exponent laws

• $x^a \cdot x^b = x^{a + b}$
• $\frac{x^a}{x^b} = x^{a - b}$
• $(x^a)^b = x^{ab}$
• $x^0 = 1$, $x^{-n} = \frac{1}{x^n}$, $x^{1/n} = \sqrt[n]{x}$
• Example: $\frac{2x^5 \cdot 3x^2}{x^4} = 6x^3$.

Logarithm laws

• $\log(ab) = \log a + \log b$
• $\log\left(\frac{a}{b}\right) = \log a - \log b$
• $\log(a^n) = n \log a$
• $\log_b b = 1$, $\log_b 1 = 0$, and $b^{\log_b x} = x$.
• Example: $\log_2 8 = 3$ because $2^3 = 8$.

Sequences

• Arithmetic: $a_n = a_1 + (n - 1)d$; sum $S_n = \frac{n}{2}(a_1 + a_n)$. For 3, 7, 11, …: the 10th term is $3 + 9 \times 4 = 39$.
• Geometric: $a_n = a_1 r^{n - 1}$; sum $S_n = \frac{a_1(1 - r^n)}{1 - r}$ for $r \neq 1$.

Coordinates

• Distance: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. From $(1, 2)$ to $(4, 6)$: $\sqrt{9 + 16} = 5$.
• Midpoint: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.

Percentages and interest

• Percentage change: $\frac{\text{new} - \text{old}}{\text{old}} \times 100\%$.
• Simple interest: $I = Prt$.
• Compound interest: $A = P\left(1 + \frac{r}{n}\right)^{nt}$.

Absolute value

• $|x| = c$ (with $c > 0$) means $x = c$ or $x = -c$.
• $|x| < c$ means $-c < x < c$; $|x| > c$ means $x < -c$ or $x > c$.

How to remember them

Formulas stick when you know where they come from. The quadratic formula comes from completing the square; the difference of two squares from expanding $(a + b)(a - b)$ and watching the middle terms cancel. Test yourself by covering one side of each line.

To practise using them, the lessons on quadratic equations, factoring polynomials and rational expressions work through examples step by step, and the quadratic calculator and factoring calculator let you check your answers.

Completing the square

• $x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2$
• Example: $x^2 + 6x + 2 = (x + 3)^2 - 9 + 2 = (x + 3)^2 - 7$, so the minimum value is $-7$, at $x = -3$.

Inequalities

• Add or subtract anything on both sides freely.
• Multiplying or dividing by a negative number reverses the sign: $-3x > 12$ gives $x < -4$.

Functions

• Composite: $f(g(x))$ — do $g$ first, then $f$. With $f(x) = 2x + 3$ and $g(x) = x^2$: $f(g(x)) = 2x^2 + 3$.
• Inverse: swap $x$ and $y$ and solve for $y$. For $f(x) = 2x + 3$: $x = 2y + 3$, so $f^{-1}(x) = \frac{x - 3}{2}$.

Systems of two equations

• Elimination: add or subtract the equations to remove one variable.
• Substitution: solve one equation for a variable and substitute into the other.
• Cramer’s rule for $ax + by = c$, $dx + ey = f$: $x = \frac{ce - bf}{ae - bd}$, $y = \frac{af - cd}{ae - bd}$, when $ae - bd \neq 0$.

Direct and inverse proportion

• Direct: $y = kx$. If $y = 12$ when $x = 3$, then $k = 4$, so $x = 5$ gives $y = 20$.
• Inverse: $y = \frac{k}{x}$. If $y = 6$ when $x = 4$, then $k = 24$, so $x = 8$ gives $y = 3$.

Rearranging formulas

Undo operations in reverse order, doing the same to both sides. From $v = u + at$: $t = \frac{v - u}{a}$. The lesson on literal equations works through more examples.

Frequently asked questions

What algebra formulas should I memorise? Start with slope, $y = mx + b$, the quadratic formula, the squares of $(a + b)$ and $(a - b)$, the difference of two squares and the exponent laws.

How do I remember the quadratic formula? Say it as a sentence — “x equals negative b, plus or minus the square root of b squared minus four a c, all over two a” — and practise deriving it once by completing the square.

Is a formula sheet allowed in exams? It depends on the exam. Some give a formula sheet, many expect you to know the basics. Check your exam’s rules.

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