The sum the zeros keep
x² − 5x + 6 has zeros 2 and 3. Their sum is 5. In ax² + bx + c, a is the number before x², so a = 1. The letter b is the number before x, so b = −5. Then −b/a = −(−5)/1 = 5. The same.
Loading lab…
Step 1 of 5
- Two quadratics, the sum checked two ways.
x² − 5x + 6: zeros 2, 3, sum 5.−b/a = 5. Both routes agree.2x² − 12x + 10: zeros 1, 5, sum 6.−b/a = 6. Both routes agree.
The rule
If `ax² + bx + c` has zeros, their sum is `−b/a`.
Here a, b and c are the coefficients. The a is in front of x². The b is in front of x. The c stands alone.
Add the zeros first. Then read a and b from the quadratic. The sum and −b/a agree whenever the quadratic has zeros. A touching curve has one zero. That zero counts twice.
| Quadratic | Zeros | Sum | −b/a |
|---|---|---|---|
x² − 5x + 6 | 2, 3 | 5 | 5 |
2x² − 12x + 10 | 1, 5 | 6 | 6 |
x² − 6x + 9 | 3, 3 | 6 | 6 |
−5x² + 40x | 0, 8 | 8 | 8 |
The curve of x² − 6x + 9 touches the axis once, at 3. Count that zero twice: 3 + 3 = 6.
Look at the last row. Here a = −5 and b = 40. So −b/a = −40/(−5) = 8, which is the sum.