Lesson 5 of 14
The decider
Show on a big screenOne number counts the real roots before you solve. Here it is called the decider.
x² − 2x − 3 = 0: decider 4 + 12 = 16, above zero. Two crossings, at −1 and 3. Step 1 of 3
x² − 2x − 3 = 0: decider4 + 12 = 16, above zero. Two crossings, at −1 and 3.x² − 4x + 4 = 0: decider16 − 16 = 0. One touch, at 2.x² + x + 1 = 0: decider1 − 4 = −3, below zero. No crossing.
In the picture, each curve is the graph of the left side. A real root is where the curve meets the x-axis.
The rule
Take ax² + bx + c = 0, with a ≠ 0. The number b² − 4ac is the discriminant. It tells you the nature of the roots. That means how many real roots, and whether they differ.
- Above zero: two different real roots.
- Exactly zero: one real root, counted twice.
- Below zero: no real root.
Counted twice means the two roots are the same.
Why? The formula has √(b² − 4ac). If b² − 4ac is 0, + and − give one root. A negative number has no real square root.
| Equation | b² − 4ac | Verdict |
|---|---|---|
x² − 2x − 3 = 0 | 4 + 12 = 16 | two different real roots |
x² − 4x + 4 = 0 | 16 − 16 = 0 | one root, counted twice |
x² + x + 1 = 0 | 1 − 4 = −3 | no real root |
2x² − 8x + 5 = 0 | 64 − 40 = 24 | two different real roots |
Tidy first. Then read a, b and c. You need no brackets and no square root.
Now try it. Work out b² − 4ac. Then tap how many roots.
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