9.3 Solving Quadratic Equations: Factoring, the Quadratic Formula & Completing the Square

Why Quadratic Equations Matter on the GED

Assessment target A.4 ("Write, manipulate, and solve quadratic equations") is smaller than A.3 in indicator count — just two indicators, A.4.a and A.4.b — but it introduces a completely different equation type that requires its own toolkit of methods. The official Assessment Guide names four acceptable solving methods for A.4.a: factoring, the quadratic formula, completing the square, and inspection. This section covers solving methods (A.4.a); section 9.4 covers writing quadratic equations from real-world context (A.4.b).

A quadratic equation is any equation that can be written in standard form: ax² + bx + c = 0, where a ≠ 0. Unlike a linear equation, which has one solution, a quadratic equation typically has two solutions (also called roots or zeros) — the two x-values where the equation equals zero.

Method 1: Factoring and the Zero Product Property

Factoring is the fastest method when the quadratic factors into whole numbers. It relies on the Zero Product Property: if two factors multiply to zero, at least one of them must equal zero.

Worked Example: Solve x² − 5x + 6 = 0.

• Find two numbers that multiply to 6 (the constant, c) and add to −5 (the coefficient, b): −2 and −3 work (−2 × −3 = 6, −2 + −3 = −5).
• Factor: (x − 2)(x − 3) = 0
• Apply the Zero Product Property: x − 2 = 0 or x − 3 = 0
• Solutions: x = 2 or x = 3

Special case — Difference of Squares: an expression of the form a² − b² always factors as (a + b)(a − b). Solve x² − 9 = 0 → (x + 3)(x − 3) = 0 → x = 3 or x = −3 (often written x = ±3).

Method 2: Inspection

"Inspection" applies to simple equations where you can isolate x² directly and take a square root. Solve x² = 16 by inspection: take the square root of both sides, remembering both the positive and negative root: x = 4 or x = −4. Forgetting the negative root when taking a square root is a frequent scoring error — x² = 16 has two solutions, not one.

Method 3: The Quadratic Formula

When a quadratic will not factor into whole numbers, the quadratic formula always works. It is provided on the official GED formula sheet, so you do not need to memorize it — but you must know how to use it:

x = (−b ± √(b² − 4ac)) / 2a

Worked Example: Solve 2x² + 3x − 2 = 0.

• Identify a = 2, b = 3, c = −2.
• Substitute: x = (−3 ± √(3² − 4(2)(−2))) / 2(2) = (−3 ± √(9 + 16)) / 4 = (−3 ± √25) / 4 = (−3 ± 5) / 4
• Two solutions: x = (−3 + 5)/4 = 0.5 and x = (−3 − 5)/4 = −2

Method 4: Completing the Square

Completing the square rewrites the equation so one side is a perfect square trinomial. Take half of b, square it, and add that value to both sides.

Worked Example: Solve x² + 6x − 7 = 0.

• Move the constant: x² + 6x = 7
• Half of 6 is 3; 3² = 9. Add 9 to both sides: x² + 6x + 9 = 16
• Left side is now a perfect square: (x + 3)² = 16
• Take the square root of both sides (keeping ±): x + 3 = ±4
• Solve: x = 1 or x = −7

Choosing a Method Quickly

On a computer-based, multiple-choice GED item, factoring or the quadratic formula are usually fastest since you can also work backward from the answer choices by substituting them into the original equation.

Common Traps to Avoid

• Forgetting the ± when taking a square root, reporting only one of the two valid solutions.
• Sign errors while factoring — finding two numbers that add to b but don't correctly multiply to c (or vice versa); always verify both conditions.
• Order-of-operations errors inside the quadratic formula, especially squaring a negative b (b² is always positive) and correctly computing 4ac's sign when c is negative.
• Not setting the equation equal to zero first — factoring and the Zero Product Property only work when one side of the equation is exactly 0.

Key Takeaways

Quadratic equations generally have two solutions, and the GED accepts four solving methods — factoring, inspection, the quadratic formula, and completing the square — with the quadratic formula provided on your formula sheet as a reliable fallback. Always double-check both solutions by substituting them back into the original equation, and never forget the ± sign whenever a square root enters your work.