Quadratic Equations • Topic 2 of 5

Factoring Method

When a quadratic factors nicely, solving is fast. Put it in standard form, factor the left side, then use the zero-product rule: if a product equals zero, at least one factor is zero. So x² + 5x + 6 = 0 factors to (x + 2)(x + 3) = 0, giving x = −2 or x = −3. For x² + bx + c, find two numbers multiplying to c and adding to b. Always factor out a common factor first. Factoring is the preferred SAT method whenever the roots are integers, because it is quicker and less error-prone than the formula.

✅ Solved examples

1. Solve x² + 5x + 6 = 0.
Factor: (x + 2)(x + 3) = 0, so x = −2 or x = −3.
2. Solve x² − 7x + 12 = 0.
Factor: (x − 3)(x − 4) = 0, so x = 3 or x = 4.
3. Solve x² − 9 = 0.
Difference of squares: (x − 3)(x + 3) = 0, so x = 3 or x = −3.
4. Solve x² + 2x = 0.
Factor x: x(x + 2) = 0, so x = 0 or x = −2.

✏️ Practice — try these, take hints as needed

1. Solve x² + 7x + 10 = 0.
Two numbers multiply to 10, add to 7.
2 and 5.
Set each factor to 0.
x = −2 or x = −5.
2. Solve x² − 5x + 6 = 0.
Multiply to 6, add to −5.
−2 and −3.
—
x = 2 or x = 3.
3. Solve x² − 16 = 0.
Difference of squares.
(x − 4)(x + 4).
—
x = 4 or x = −4.
4. Solve x² − 3x = 0.
Factor out x.
x(x − 3) = 0.
—
x = 0 or x = 3.
5. Solve x² + x − 12 = 0.
Multiply to −12, add to 1.
4 and −3.
—
x = −4 or x = 3.

📝 Topic test — 8 questions

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