Algebra Cheat Sheet: Essential Formulas & Concepts for Quick Reference

This Algebra cheat sheet is your quick guide to mastering key concepts, formulas, and techniques. It's a compact reference for the essentials, from solving linear equations to working with exponents and factoring quadratics. Perfect for quick review or exam prep.

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1. Basic Properties and Facts:

Exponent Properties

• $1^p = 1 \ \ \ \ \ \ \ \ \ \ a^1 = a \ \ \ \ \ \ \ \ \ \ a^0 = 1, a \ne 0 \ \ \ \ \ \ \ \ \ \ 0^a = 0, a \ne 0$

  
  

• $p^n p^m = p^{n+m} \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space (pq)^n = p^n q^n$

  
  

• $(p^n)^m = p^{nm} \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space p^0 = 1, p \neq 0$

  
  

• $\dfrac{p^n}{p^m} = p^{n-m} = \dfrac{1}{p^{m-n}} \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \left(\dfrac{p}{q}\right)^n = \dfrac{p^n}{q^n}$

  
  

• $p^{(\dfrac{n}{m})} = \left(p^{\dfrac{1}{m}}\right)^n = \left(p^n\right)^{\frac{1}{m}} \space \space \space \space \space \space \space \space \space \dfrac{1}{p^{-n}} = p^n$

  
  

• $\left(\dfrac{p}{q}\right)^{-n} = \left(\dfrac{q}{p}\right)^n = \dfrac{q^n}{p^n} \space \space \space \space \space \space \space \space \space \space \space \space \space \space p^{-n} = \dfrac{1}{p^n}$

Number Rules

• $a.0 = 0$

• $1.a = a$

Arithmetic Operations

• $-(-a) = a$
  
• $(-1)(a \plusmn b) = -a ∓ b$
  
• $ab + ac = a(b + c)$
  
• $(a+b)(c+d) = ac + ad + bc + bd$
  
• $(a)(b+c)(d+e) = and + abe + acd + ace$
  

Fraction Rules

• $\dfrac{0}{a} = 0, a \ne 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \dfrac{a}{1} = a \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \dfrac{a}{a} = 1$

  
  

• $(a)^{-1} = \dfrac{1}{a} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\dfrac{a}{b})^{-1} = \dfrac{1}{\dfrac{a}{b}} = \dfrac{b}{a}$

  
  

• $a({\dfrac{b}{c}}) = \dfrac{ab}{c}$

  
  

• $\dfrac{(\dfrac{a}{b})}c = \dfrac{a}{bc}$

  
  

• $\dfrac{a}{(\dfrac{b}{c})} = \dfrac{ac}{b}$

  
  

• $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}$

Operation on Fractions Calculators

• $\dfrac{a}{b} - \dfrac{c}{d} = \dfrac{ad - bc}{bd}$

  
  

• $\dfrac{a - b}{c - d} = \dfrac{b - a}{d - c}$

  
  

• $\dfrac{a + b}{c} = \dfrac{a}{c} + \dfrac{b}{c}$

  
  

• $\dfrac{ab + ac}{a} = b + c, a \neq 0$

  
  

• $\dfrac{(\dfrac{a}{b})}{(\dfrac{c}{d})} = \dfrac{ad}{bc}$

Properties of Radicals

• $\sqrt{1} = 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sqrt{0} = 0$

  
  

• $\sqrt[n]{a} = a^{\dfrac{1}{n}} \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \sqrt[n]{ab} = \sqrt[n]{a} \sqrt[n]{b}$

  
  

• $\sqrt[m]{\sqrt[n]{a}} = \sqrt[nm]{a} \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}, a, b \geq 0$

  
  

• $\sqrt[n]{a^n} = a$ if $n$ is odd $\space \space \space \space \space$ $\sqrt[n]{a^n} = |a|$ if $n$ is even

Properties of Absolute Value

• $|ax| = a|x|, a \geq 0$

  
  

• $|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$

  
  

• $|a| \geq 0 \space \space \space \space \space \space \space \space \space \space \space \space \space \space |-a| = |a|$

  
  

• $|ab| = |a||b| \space \space \space \space \space \space \space \space \space \left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|}$

  
  

• $|a + b| \leqslant |a| + |b| \space \space \space \space \space$ (Triangle Inequality)

Properties of Inequalities

• If $a < b$, then $a + c < b + c$ and $a - c < b - c$

  
  

• If $a < b$ and $c > 0$, then $ac < bc$ and $\dfrac{a}{c} < \dfrac{b}{c}$

  
  

• If $a < b$ and $c < 0$, then $ac > bc$ and $\dfrac{a}{c} > \dfrac{b}{c}$

Distance Formula

Distance between two points Calculator
If $A = (x_1, y_1)$ and $B = (x_2, y_2)$ are two points the distance between them is

$d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Complex Numbers

Complex Number Calculators

• $i = \sqrt{-1}, i^2 = -1 \space \space \space \space \space \space \space \sqrt{-a} = i\sqrt{a}, a \geq 0$

  
  

• $(a + bi) + (c + di) = (a + c) + (b + d)i$

  
  

• $(a + bi) - (c + di) = (a - c) + (b - d)i$

  
  

• $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

  
  

• $(a + bi)(a - bi) = a^2 + b^2$

  
  

• $|a + bi| = \sqrt{a^2 + b^2}$ (Complex Modulus)

  
  

• $\overline{a + bi} = a - bi$ (Complex Conjugate)

  
  

• $(a + bi)(a + bi) = |a + bi|^2$

2. Logarithms and Log Properties:

Definition

$y = \log_b(x) \iff$ $x = b^y$

Example: $\log_5(125) = 3$ because $5^3 = 125$

Special Logarithms

• $\ln(x) = \log_e(x)$ .......called as (natural log)

• $\log(x) = \log_{10}(x)$ ..... called as(common log)

where $e = 2.718281828...$

Logarithm Formula Sheet

Logarithm Properties

• $\log_b(b) = 1 \space \space \space \space \space \space \space \space \space \log_b(1) = 0$

  
  

• $\log_b(b^x) = x \space \space \space \space \space \space \space b^{\log_b(x)} = x$

  
  

• $\log_b(x^r) = r \space \log_b(x)$

  
  

• $\log_b(xy) = \log_b(x) + \log_b(y)$

  
  

• $\log_b\left(\dfrac{x}{y}\right) = \log_b(x) - \log_b(y)$

  
  

• The domain of $\log_b(x)$ is $x > 0$

Undefined

• $0^0 = \text{Undefined}$

  
  

• $\dfrac{x}{0} = \text{Undefined}$

  
  

• $\log_1(p) = \text{Undefined}$

  
  

• $\log_q(p) = \text{Undefined}, p \leq 0$

  
  

• $\log_p(q) = \text{Undefined}, q \leq 0$

3. Factoring and Solving:

Factoring Formulas

• $x^2 - a^2 = (x + a)(x - a)$

  
  

• $x^2 + 2ax + a^2 = (x + a)^2$

  
  

• $x^2 - 2ax + a^2 = (x - a)^2$

  
  

• $x^2 + (a + b)x + ab = (x + a)(x + b)$

  
  

• $x^3 + 3ax^2 + 3a^2x + a^3 = (x + a)^3$

  
  

• $x^3 - 3ax^2 + 3a^2x - a^3 = (x - a)^3$

  
  

• $x^3 + a^3 = (x + a)(x^2 - ax + a^2)$

  
  

• $x^3 - a^3 = (x - a)(x^2 + ax + a^2)$

  
  

• $x^{2n} - a^{2n} = (x^n - a^n)(x^n + a^n)$

  
  

If $n$ is odd then,

• $x^n - a^n = (x - a)(x^{n-1} + ax^{n-2} + \cdots + a^{n-1})$

  
  

• $x^n + a^n = (x + a)(x^{n-1} - ax^{n-2} + a^2x^{n-3} - \cdots + a^{n-1})$

Quadratic Formula

Quadratic Equation Calculator
Solve $ax^2 + bx + c = 0$, $a \neq 0$, use

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

If $b^2 - 4ac > 0$ – Two real unequal solns.

If $b^2 - 4ac = 0$ - Repeated real solution.

If $b^2 - 4ac < 0$ - Two complex solutions.

Square Root Property

If $x^2 = p$ then $x = \pm \sqrt{p}$

Absolute Value Equations/Inequalities

If $b$ is a positive number

$|p| = b \Rightarrow p = -b \text{ or } p = b$

$|p| < b \Rightarrow -b < p < b$

$|p| > b \Rightarrow p < -b \text{ or } p > b$

4. Completing the Square:

Solve $4x^2 - 12x - 20 = 0$

(1) Divide by the coefficient of $x^2$ i.e. 4 by whole equation = $x^2 - 3x - 5 = 0$

(2) Move the constant to the other side: $x^2 - 3x = 5$

(3) Take half the coefficient of $x$, square it, and add it to both sides

$x^2 - 3x + \left(-\dfrac{3}{2}\right)^2 = 5 + \left(-\dfrac{3}{2}\right)^2 = 5 + \dfrac{9}{4} = \dfrac{29}{4}$

(4) Factor the left side $\left(x - \dfrac{3}{2}\right)^2 = \dfrac{29}{4}$

(5) Use the Square Root Property
$x - \dfrac{3}{2} = \pm \dfrac{\sqrt{29}}{4} = \pm \dfrac{\sqrt{29}}{2}$

(6) Solve for $x$: $x = \dfrac{3}{2} \pm \dfrac{\sqrt{29}}{2}$

5. Functions and Graphs:

Constant Function

$y = a$ or $f(x) = a$

Graph is a horizontal line passing through the point $(0, a)$.

Line/Linear Function

$y = mx + b$ or $f(x) = mx + b$

Graph is a line with point $(0, b)$ and slope $m$.

Slope

Slope of the line containing the two points $(x_1, y_1)$ and $(x_2, y_2)$ is

$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{\Delta y }{\Delta x} = \dfrac{\text{distance rise}}{\text{distance run}}$

Slope-Intercept Form

The equation of the line with slope $m$ and $y$-intercept $(0, b)$ is

$y = mx + b$

Point-Slope Form

The equation of the line with slope $m$ and passing through the point $(x_1, y_1)$ is

$y = y_1 + m(x - x_1)$

Parabola/Quadratic Function

$y = a(x - h)^2 + k; \space \space \space \space f(x) = a(x - h)^2 + k$

The graph is a parabola that opens up if $a > 0$ or down if $a < 0$ and has a vertex at $(h, k)$.

Parabola/Quadratic Function

$y = ax^2 + bx + c; \space \space \space \space \space \space f(x) = ax^2 + bx + c$

The graph is a parabola that opens up if $a > 0$ or down if $a < 0$ and has a vertex at $\left(-\dfrac{b}{2a}, f\left(-\dfrac{b}{2a}\right)\right)$.

Parabola/Quadratic Function

$x = ay^2 + by + c \space \space \space \space \space \space \space g(y) = ay^2 + by + c$

The graph is a parabola that opens right if $a > 0$ or left if $a < 0$ and has a vertex at $\left(g\left(-\dfrac{b}{2a}\right), -\dfrac{b}{2a}\right)$.

Circle

$(x - h)^2 + (y - k)^2 = r^2$

Graph is a circle with radius $r$ and center $(h, k)$.

Ellipse

$\dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2} = 1$

Graph is an ellipse with center $(h, k)$, with vertices $a$ units right/left from the center and vertices $b$ units up/down from the center.

Hyperbola

$\dfrac{(x - h)^2}{a^2} - \dfrac{(y - k)^2}{b^2} = 1$

Graph is a hyperbola that opens left and right, has a center at $(h, k)$, vertices $a$ units left/right of the center, and asymptotes that pass through the center with slope $\pm \dfrac{b}{a}$.

Hyperbola

$\dfrac{(y - k)^2}{b^2} - \dfrac{(x - h)^2}{a^2} = 1$

Graph is a hyperbola that opens up and down, has a center at $(h, k)$, vertices $b$ units up/down from the center, and asymptotes that pass through the center with slope $\pm \dfrac{b}{a}$.

6. Common Algebraic Errors:

Error $\space \space \space \space \space \space \space\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Reason/Correct/Justification/Example

• $\dfrac{2}{0} \neq 0$ and $\dfrac{2}{0} \neq 2$ $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Division by zero is undefined!

  
  

• $-3^2 \neq 9 \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space -3^2 = -9$, but $(-3)^2 = 9$ Pay attention to the parenthesis

  
  

• $(x^2)^3 \neq x^5 \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space (x^2)^3 = x^2 \space x^2 \space x^2 = x^6$

  
  

• $\dfrac{a}{b+c} \neq \dfrac{a}{b} + \dfrac{a}{c} \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \dfrac{1}{2} = \dfrac{1}{1+1} \neq \dfrac{1}{1} + \dfrac{1}{1} = 2$

  
  

• $\dfrac{1}{x^2 x^3} \neq {x}^{-2} + {x}^{-3} \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ A more complex version of the previous error.

  
  

• $\dfrac{\cancel{a}+bx}{\cancel{a}} \neq 1 + bx \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \dfrac{a+bx}{a} = \dfrac{a}{a} + \dfrac{bx}{a} = 1 + \dfrac{bx}{a} \space$ Beware of incorrect cancelling!

  
  

• $-a(x - 1) \neq -ax - a \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space -a(x - 1) = -ax + a \space$ Make sure you distribute the “-”

  
  

• $(x + a)^2 \neq x^2 + a^2 \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space (x + a)^2 = (x + a)(x + a) = x^2 + 2ax + a^2$

  
  

• $\sqrt{x^2 + a^2} \neq x + a \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space 5= \sqrt{25} = \sqrt{3^2 + 4^2} \neq \sqrt{3^2} + \sqrt{4^2} = 3 + 4 = 7$

  
  

• $\sqrt{x + a} \neq \sqrt{x} + \sqrt{a} a \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ See previous error

  
  

• $(x + a)^n \neq x^n + a^n$
  and $\sqrt[n]{x + a} \neq \sqrt[n]{x} + \sqrt[n]{a} \space \space \space \space \space \space \space \space \space \space \space \space$ More general versions of previous three errors.

  
  

• $2(x+1)^2 \neq (2x+2)^2 \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space 2(x+1)^2 = 2(x^2 + 2x + 1) = 2x^2 + 4x + 2$

$\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space (2x+2)^2 = 4x^2 + 8x + 4$ $\space \space \space \space \space \space \space \space \space \space$ Square first, then distribute!

• $(2x+2)^2 \neq 2(x+1)^2$ $\space \space \space \space \space \space \space \space \space \space \space \space \space$ See the previous example. You cannot factor out a constant if there is a power on the parenthesis!

  
  

• $\sqrt{-x^2 + a^2} \neq -\sqrt{x^2 + a^2} \space \space \space \space \space \space \space \space \space \sqrt{-x^2 + a^2} = (-x^2 + a^2)^{\dfrac{1}{2}} \space \space \space \space \space \space$ Now see the previous error.

  
  

• $\dfrac{a}{(\dfrac{b}{c})} \neq \dfrac{ab}{c} \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \dfrac{a}{(\dfrac{b}{c})} = \dfrac{(\dfrac{a}{1})}{(\dfrac{b}{c})} = (\dfrac{a}{1}) (\dfrac{c}{b}) = \dfrac{ac}{b}$

  
  

• $\frac{(\dfrac{a}{b})}{c} \neq \dfrac{ac}{b} \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \dfrac{(\dfrac{a}{b})}{c} = \dfrac{(\dfrac{a}{b})}{(\dfrac{c}{1})} = (\dfrac{a}{b})(\dfrac{1}{c}) = \dfrac{a}{bc}$

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