What the relations tell you first
No solving yet. What do the sum S and the product P already say?
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- A number line, split at 0.
Pnegative: zeros on opposite sides. Example: −5, 2.Ppositive,Snegative, zeros exist: both negative. Example: −3, −4.Ppositive,Spositive, zeros exist: both positive. Example: 4, 5.
The rule
If zeros exist, a positive product means they share a side. A negative product means they sit on opposite sides. A side means the positive side or the negative side of 0. Read P first. Then read S. When the zeros share a side, S shows which. A positive S means both are positive. A negative S means both are negative.
| Quadratic | P | S | What it says | Zeros |
|---|---|---|---|---|
x² + 3x − 10 | −10 | −3 | opposite sides; the negative zero is farther from 0 | −5, 2 |
x² + 7x + 12 | 12 | −7 | same side, both negative | −3, −4 |
x² − 9x + 20 | 20 | 9 | same side, both positive | 4, 5 |
None of these rows needed a split.
One known zero gives the other. Take 3x² − 10x + 8. At x = 2 the value is 12 − 20 + 8 = 0, so 2 is a zero. The product of the zeros is c/a = 8/3. So the other zero is (8/3) ÷ 2 = 4/3. The sum agrees: 2 + 4/3 = 10/3.
A known zero also finds an unknown number. If 3 is a zero of x² − 7x + c, use x = 3: 9 − 21 + c = 0. So c = 12.