Intermediate Algebra and Coordinate Geometry

Quadratic Equations

Factoring

$x^2 + 5x + 6 = 0$ $(x + 2)(x + 3) = 0$ $x = -2 \text{ or } x = -3$

Quadratic Formula

For $ax^2 + bx + c = 0$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Completing the Square

$x^2 + 6x = 7$ $x^2 + 6x + 9 = 7 + 9$ $(x + 3)^2 = 16$ $x + 3 = \pm 4$ $x = 1 \text{ or } x = -7$

Inequalities

Solving: $2x - 5 < 7$ $2x < 12$ $x < 6$

Important: When multiplying/dividing by negative, flip the sign! $-3x > 6 \quad \Rightarrow \quad x < -2$

Systems of Equations

Substitution Method

$\begin{cases} y = 2x + 1 \\ 3x + y = 11 \end{cases}$

Substitute: $3x + (2x + 1) = 11$ $5x = 10 \quad \Rightarrow \quad x = 2, y = 5$

Elimination Method

$\begin{cases} 2x + 3y = 12 \\ 2x - y = 4 \end{cases}$

Subtract: $4y = 8 \quad \Rightarrow \quad y = 2$

Coordinate Geometry

Distance Formula

Between $(x_1, y_1)$ and $(x_2, y_2)$: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Midpoint Formula

$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

Slope

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}}$

Special cases:

• Horizontal line: slope = 0
• Vertical line: slope = undefined
• Parallel lines: same slope
• Perpendicular lines: slopes are negative reciprocals

Slope-Intercept Form

$y = mx + b$

• $m$ = slope
• $b$ = y-intercept

ACT Tips

• Quadratics: Try factoring first (fastest), then formula
• Graph questions: Sketch if not provided
• Systems: Use substitution when one equation is already solved
• Calculator: Can graph to find intersections