Solving Quadratic Equations – Review Flashcards

A set of flashcards reviewing definitions and solution techniques for quadratic equations, including factoring, disguised quadratics, completing the square, and the quadratic formula.

What is a quadratic equation?

An equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

What are the steps to solve a quadratic equation by factoring?

1) Write the equation in standard form ax² + bx + c = 0. 2) Factor the quadratic expression. 3) Set each factor equal to 0 and solve for x.

Solve by factoring: x² + 2x – 8 = 0.

(x + 4)(x – 2) = 0 ⟹ x = –4 or x = 2.

What are “disguised quadratics” and how are they solved?

Equations like x⁴ + … or x⁸ + … that can be rewritten as a quadratic in a new variable (e.g., let y = x²). Substitute, solve the quadratic in y, then back-substitute for x.

Using substitution y = x⁴, how do you solve x⁸ + 3x⁴ – 40 = 0?

x⁸ + 3x⁴ – 40 = 0 → y² + 3y – 40 = 0 → (y + 8)(y – 5) = 0 → y = –8 (reject for real x) or y = 5 → x⁴ = 5 → x = ±√5 (two real solutions).

What does “completing the square” mean?

Rewriting a quadratic ax² + bx + c in the form a(x + h)² + k by adding and subtracting the square of (b/2a), making it easier to solve or graph.

Complete the square for x² + 4x + 1.

x² + 4x + 1 = (x + 2)² – 3.

Find the exact solutions of 2x² – 8x + 3 = 0 by completing the square.

Factor 2: 2(x² – 4x) + 3 = 0 → 2[(x – 2)² – 4] + 3 = 0 → (x – 2)² = 5/2 → x = 2 ± √(5/2).

When is it best to use the quadratic formula?

When factoring is difficult or impossible and completing the square is cumbersome.

What is the quadratic formula?

x = [–b ± √(b² – 4ac)] / (2a).

Solve x² – 3x + 1 = 0 exactly using the quadratic formula.

a = 1, b = –3, c = 1 → x = [3 ± √(9 – 4)] / 2 = (3 ± √5)/2.

What is the solution to x² + 12 = 0?

x² = –12 → x = ±2√3 i (no real solutions).