Building it from its zeros
Two numbers add to 9 and multiply to 20. Which quadratic has them as zeros?
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Step 1 of 5
- Zeros given: 4 and 5.
- Sum:
S = 4 + 5 = 9. - Product:
P = 4 × 5 = 20. - Quadratic:
x² − 9x + 20. - Check:
x = 4gives16 − 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.
| Zeros | S | P | Quadratic |
|---|---|---|---|
| 3, 7 | 10 | 21 | x² − 10x + 21 |
1/2, 4 | 9/2 | 2 | x² − (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.