🔤 Algebra — Formulas

Essential formulas and shortcuts for working with algebra in MATHCOUNTS.

Basic Algebraic Formulas

Order of Operations

PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction Example: $2 + 3 \times 4^2 = 2 + 3 \times 16 = 2 + 48 = 50$

Properties of Equality

Addition: If $a = b$, then $a + c = b + c$ Subtraction: If $a = b$, then $a - c = b - c$ Multiplication: If $a = b$, then $ac = bc$ Division: If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$

Properties of Operations

Commutative: $a + b = b + a$, $ab = ba$ Associative: $(a + b) + c = a + (b + c)$, $(ab)c = a(bc)$ Distributive: $a(b + c) = ab + ac$, $a(b - c) = ab - ac$

Linear Equations

One-Step Equations

Addition: $x + a = b$ → $x = b - a$ Subtraction: $x - a = b$ → $x = b + a$ Multiplication: $ax = b$ → $x = \frac{b}{a}$ Division: $\frac{x}{a} = b$ → $x = ab$

Two-Step Equations

General form: $ax + b = c$ → $x = \frac{c - b}{a}$ Example: $2x + 3 = 11$ → $x = \frac{11 - 3}{2} = 4$

Multi-Step Equations

Method: Simplify each side, then solve Example: $3(x + 2) - 4 = 8$

  1. Distribute: $3x + 6 - 4 = 8$
  2. Combine: $3x + 2 = 8$
  3. Solve: $x = 2$

Systems of Equations

Substitution Method

Step 1: Solve one equation for one variable Step 2: Substitute into other equation Step 3: Solve for remaining variable Step 4: Find other variable

Example: Solve $\begin{cases} x + y = 5 \ 2x - y = 1 \end{cases}$

  1. $y = 5 - x$
  2. $2x - (5 - x) = 1$
  3. $3x = 6$, so $x = 2$
  4. $y = 5 - 2 = 3$

Elimination Method

Step 1: Align like terms Step 2: Add or subtract equations to eliminate one variable Step 3: Solve for remaining variable Step 4: Find other variable

Example: Solve $\begin{cases} x + y = 5 \ 2x - y = 1 \end{cases}$

  1. Already aligned
  2. Add: $3x = 6$
  3. $x = 2$
  4. $y = 3$

Quadratic Equations

Standard Form

General form: $ax^2 + bx + c = 0$ where $a \neq 0$

Factoring

Difference of squares: $a^2 - b^2 = (a + b)(a - b)$ Perfect square trinomial: $a^2 + 2ab + b^2 = (a + b)^2$ Perfect square trinomial: $a^2 - 2ab + b^2 = (a - b)^2$ General trinomial: $x^2 + (a + b)x + ab = (x + a)(x + b)$

Examples:

  • $x^2 - 9 = (x + 3)(x - 3)$
  • $x^2 + 6x + 9 = (x + 3)^2$
  • $x^2 - 6x + 9 = (x - 3)^2$
  • $x^2 + 5x + 6 = (x + 2)(x + 3)$

Quadratic Formula

Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Discriminant: $b^2 - 4ac$

  • If $b^2 - 4ac > 0$: Two real solutions
  • If $b^2 - 4ac = 0$: One real solution
  • If $b^2 - 4ac < 0$: No real solutions

Example: Solve $x^2 - 5x + 6 = 0$

  • $a = 1$, $b = -5$, $c = 6$
  • $x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2}$
  • Solutions: $x = 3$ or $x = 2$

Completing the Square

Method: Rewrite as $(x - h)^2 = k$ Example: Solve $x^2 + 6x + 5 = 0$

  1. Move constant: $x^2 + 6x = -5$
  2. Add $(\frac{6}{2})^2 = 9$: $x^2 + 6x + 9 = 4$
  3. Factor: $(x + 3)^2 = 4$
  4. Solve: $x + 3 = \pm 2$, so $x = -1$ or $x = -5$

Inequalities

Linear Inequalities

Addition: If $a < b$, then $a + c < b + c$ Subtraction: If $a < b$, then $a - c < b - c$ Multiplication: If $a < b$ and $c > 0$, then $ac < bc$ Division: If $a < b$ and $c > 0$, then $\frac{a}{c} < \frac{b}{c}$ Negative multiplication: If $a < b$ and $c < 0$, then $ac > bc$ Negative division: If $a < b$ and $c < 0$, then $\frac{a}{c} > \frac{b}{c}$

Compound Inequalities

And: $a < x < b$ means $a < x$ and $x < b$ Or: $x < a$ or $x > b$ means either condition is true

Examples:

  • $2 < x < 5$: All numbers between 2 and 5
  • $x < 2$ or $x > 5$: All numbers less than 2 or greater than 5

Functions

Function Notation

Definition: $f(x) = 2x + 3$ means “f of x equals 2x plus 3” Evaluation: $f(4) = 2(4) + 3 = 11$ Domain: Set of all possible inputs Range: Set of all possible outputs

Function Types

Linear: $f(x) = mx + b$ where $m$ is slope, $b$ is y-intercept Quadratic: $f(x) = ax^2 + bx + c$ where $a \neq 0$ Absolute value: $f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}$ Square root: $f(x) = \sqrt{x}$ where $x \geq 0$

Function Operations

Addition: $(f + g)(x) = f(x) + g(x)$ Subtraction: $(f - g)(x) = f(x) - g(x)$ Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$ Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ where $g(x) \neq 0$ Composition: $(f \circ g)(x) = f(g(x))$

Example: If $f(x) = 2x + 3$ and $g(x) = x^2$, then $(f \circ g)(x) = f(x^2) = 2x^2 + 3$

Sequences and Series

Arithmetic Sequences

General term: $a_n = a_1 + (n-1)d$ where $d$ is common difference Sum of first n terms: $S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$

Example: 2, 5, 8, 11, … has $d = 3$

  • $a_n = 2 + (n-1)(3) = 3n - 1$
  • $S_{10} = \frac{10}{2}(2 + 29) = 5(31) = 155$

Geometric Sequences

General term: $a_n = a_1 \cdot r^{n-1}$ where $r$ is common ratio Sum of first n terms: $S_n = \frac{a_1(1 - r^n)}{1 - r}$ where $r \neq 1$

Example: 2, 6, 18, 54, … has $r = 3$

  • $a_n = 2 \cdot 3^{n-1}$
  • $S_5 = \frac{2(1 - 3^5)}{1 - 3} = \frac{2(1 - 243)}{-2} = 242$

Fibonacci Sequence

Definition: $F_n = F_{n-1} + F_{n-2}$ with $F_1 = 1$, $F_2 = 1$ First few terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …

Word Problem Formulas

Age Problems

Current age: $x$ Age in n years: $x + n$ Age n years ago: $x - n$

Example: “John is 5 years older than Mary. In 3 years, John will be twice as old as Mary.”

  • Let $x$ = Mary’s age now
  • John’s age now: $x + 5$
  • In 3 years: Mary = $x + 3$, John = $x + 8$
  • Equation: $x + 8 = 2(x + 3)$
  • Solution: $x = 2$

Distance Problems

Basic formula: $d = rt$ where $d$ = distance, $r$ = rate, $t$ = time Same direction: $d_1 - d_2 = (r_1 - r_2)t$ Opposite direction: $d_1 + d_2 = (r_1 + r_2)t$

Example: Two cars start 100 miles apart, one goes 60 mph, other 40 mph

  • Relative speed: $60 + 40 = 100$ mph
  • Time to meet: $\frac{100}{100} = 1$ hour

Mixture Problems

Weighted average: $\frac{a_1 \times p_1 + a_2 \times p_2 + … + a_n \times p_n}{a_1 + a_2 + … + a_n}$

Example: Mix 2 liters of 20% solution with 3 liters of 80% solution

  • $\frac{2 \times 0.20 + 3 \times 0.80}{2 + 3} = \frac{0.4 + 2.4}{5} = 0.56 = 56%$

Work Problems

Individual rate: $r = \frac{\text{work}}{\text{time}}$ Combined rate: $r_{\text{combined}} = r_1 + r_2 + … + r_n$ Time to complete: $t = \frac{\text{work}}{r_{\text{combined}}}$

Example: Pipe A fills tank in 3 hours, Pipe B in 6 hours

  • Combined rate: $\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$ tank/hour
  • Time together: 2 hours

Mental Math Shortcuts

Common Patterns

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

Examples:

  • $(x + 3)^2 = x^2 + 6x + 9$
  • $(x - 3)^2 = x^2 - 6x + 9$
  • $(x + 3)(x - 3) = x^2 - 9$

Quick Substitution

When $x = 2$:

  • $x^2 = 4$
  • $x^3 = 8$
  • $2x = 4$
  • $3x = 6$

When $x = 3$:

  • $x^2 = 9$
  • $x^3 = 27$
  • $2x = 6$
  • $3x = 9$

Common Fractions

$\frac{1}{2} = 0.5$ $\frac{1}{3} = 0.333…$ $\frac{2}{3} = 0.666…$ $\frac{1}{4} = 0.25$ $\frac{3}{4} = 0.75$

Common Applications

Business Problems

Profit: Profit = Revenue - Cost Markup: Selling price = Cost + (Markup % × Cost) Discount: Sale price = Original price - (Discount % × Original price)

Science Problems

Density: Density = $\frac{\text{Mass}}{\text{Volume}}$ Pressure: Pressure = $\frac{\text{Force}}{\text{Area}}$ Speed: Speed = $\frac{\text{Distance}}{\text{Time}}$

Geometry Problems

Perimeter: Sum of all sides Area: Length × Width (for rectangles) Volume: Length × Width × Height (for rectangular prisms)

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