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Solving quadratics by factoring worksheets

A quadratic set equal to zero is solved in two moves: factor it, then set each bracket to zero. These free printable worksheets give eight equations like x² − 11x + 28 = 0, and the answer key writes the brackets, the two small equations and both roots.

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Solving quadratics by factoring
Solving quadratics by factoring worksheet — Factor first, roots to ±20, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

Factoring comes first because it is the step that makes the rest possible. x² + 4x − 165 = 0 cannot be solved by moving numbers from side to side; (x − 11)(x + 15) = 0 can, because a product is zero only when one of its factors is. The search is the familiar one — two numbers that multiply to −165 and add to 4 — but here the brackets are not the answer. The answer is what the brackets force x to be.

Why this sheet works

The key sets that second half out in full: (x − 11)(x + 15) = 0, then x − 11 = 0 or x + 15 = 0, then each one solved, then x = 11 or x = −15. The middle line is where the marks go, because the numbers inside the brackets are the roots with their signs turned over. A student who factors correctly and then copies −11 and 15 out of the brackets has done the hard part and still written the wrong answer.

How this one works

x² − 11x + 28 = 0

x² − 11x + 28 = 0
(x − 4)(x − 7) = 0
(x − 4)(x − 7) = 0
x − 4 = 0orx − 7 = 0
x − 4 = 0→x = 4
x − 7 = 0→x = 7
x = 4orx = 7
  1. Factor the left side first: two numbers that multiply to 28 and add to −11 — −4 × −7 = 28 and −4 + −7 = −11.
  2. If two things multiply to zero, one of them must be zero. So either (x − 4) is zero, or (x − 7) is zero.
  3. Solve each one. x − 4 = 0 gives x = 4, and x − 7 = 0 gives x = 7. Notice the sign turns over: a − in the bracket gives a positive answer.
  4. So x = 4 or x = 7. Either one makes the equation true.

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