Precalc Test 1

Linear Equation

An equation whose graph is a straight line; variables appear only to the first power.

Standard Form

Ax + By = C

Slope–Intercept Form

y = mx + b, where m is slope and b is the y-intercept.

Point–Slope Form

y - y₁ = m(x - x₁)

Slope Formula

m = (y₂ - y₁) / (x₂ - x₁)

Parallel Lines

Lines with equal slopes (m₁ = m₂).

Perpendicular Lines

Lines whose slopes multiply to -1 (m₁ • m₂ = -1).

Literal Equation

An equation involving multiple variables; solved for one variable in terms of the others.

Solving Literal Equations

Use inverse operations to isolate the desired variable while maintaining balance.

Example – Area Formula

From A = bh, solve for h: h = A / b.

Example – Volume Formula

From V = lwh, solve for w: w = V / (lh).

Standard Quadratic Form

ax² + bx + c = 0.

Solving by Factoring

Factor the quadratic expression and set each factor equal to 0.

Square Root Property

If x² = k, then x = ±√k.

Completing the Square

Make a = 1, add (b/2)² to both sides, factor, and take the square root.

Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a.

Discriminant

b² - 4ac; determines number and type of roots.

Discriminant > 0

Two distinct real solutions.

Discriminant = 0

One real solution (repeated root).

Discriminant < 0

No real solutions (two complex roots).

Rational Equation

An equation containing fractions with variables in the denominator.

Solving Rational Equations

Multiply by the least common denominator (LCD) to eliminate fractions.

Least Common Denominator (LCD)

The smallest expression that all denominators divide into evenly.

Extraneous Solution

A solution that makes a denominator 0 and must be excluded.

Radical Equation

An equation that contains a variable inside a square root or other root.

Solving Radical Equations

Isolate the radical, square both sides, solve, and check for extraneous roots.

Quadratic in Form

An equation that can be rewritten as a quadratic using substitution, e.g. x⁴ - 5x² + 4 = 0 → let u = x².

Substitution Method

Replace a power of the variable with a single letter, solve, then back-substitute.

Distance Formula

d = √((x₂ - x₁)² + (y₂ - y₁)²).

Midpoint Formula

M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Slope Formula

m = (y₂ - y₁) / (x₂ - x₁).

Slope–Intercept Equation

y = mx + b.

Point–Slope Equation

y - y₁ = m(x - x₁).

Standard Form Equation

Ax + By = C.

Positive Slope

Line rises left to right.

Negative Slope

Line falls left to right.

Zero Slope

Horizontal line.

Undefined Slope

Vertical line.