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Lesson 6 of 13

Building it from its zeros

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7 min

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Two numbers add to 9 and multiply to 20. Which quadratic has them as zeros?

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Zeros given: 4 and 5.
Step 1 of 5
  1. Zeros given: 4 and 5.
  2. Sum: S = 4 + 5 = 9.
  3. Product: P = 4 × 5 = 20.
  4. Quadratic: x² − 9x + 20.
  5. Check: x = 4 gives 16 − 36 + 20 = 0.

The rule

The numbers are 4 and 5. So (x − 4)(x − 5) has them as zeros. Multiply: (x − 4)(x − 5) = x² − 5x − 4x + 20 = x² − 9x + 20. The same steps work for any two zeros. That is the recipe.

Zeros with sum `S` and product `P` give `x² − Sx + P`.

Watch the middle sign. It is minus the sum.

The recipe on two more pairs of zeros
ZerosSPQuadratic
3, 71021x² − 10x + 21
1/2, 49/22x² − (9/2)x + 2

The recipe works with fractions too. Multiply every term by 2 to clear them: 2x² − 9x + 4.

Let k be any number that is not 0. Multiply every term by k. The zeros stay the same. So many quadratics have these two zeros. For example, 2x² − 18x + 40 has zeros 4 and 5.

Try the lab to finish this lesson.