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ALGEBRA · QUADRATICS

Factoring Quadratics

Rewrite a quadratic as a product so you can read the roots straight off.

14 min read 3 practice questions
A quadratic has the form ax2+bx+cax^2 + bx + c. Factoring rewrites it as a product, like (x+p)(x+q)(x + p)(x + q). Because a product is zero only when a factor is zero, factoring turns one hard equation into two easy ones.
x2+bx+c=(x+p)(x+q)wherep+q=b,  pq=cx^2 + bx + c = (x + p)(x + q) \quad\text{where}\quad p + q = b,\; pq = c

Worked example · Find two numbers

Solve x2+5x+6=0x^2 + 5x + 6 = 0.
  1. 1
    Find two numbers that multiply to 66 and add to 55: 22 and 33.
  2. 2
    Factor: (x+2)(x+3)=0(x + 2)(x + 3) = 0.
  3. 3
    Zero product property: x=−2x = -2 or x=−3x = -3.

Worked example · Difference of squares

Factor x2−49x^2 - 49.
  1. 1
    Recognize a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b) with a=xa = x, b=7b = 7.
  2. 2
    So x2−49=(x−7)(x+7)x^2 - 49 = (x - 7)(x + 7).

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

1
Factor x2+7x+12x^2 + 7x + 12.
2
What are the solutions of x2−x−20=0x^2 - x - 20 = 0?
3
x2−81=(x−a)(x+a)x^2 - 81 = (x - a)(x + a) with a>0a > 0. Find aa.

Numeric answers accept decimals or fractions like 3/4.