Two roots, two kinds
Two equations both have a decider, b² − 4ac, above zero. Their roots look very different.
x² − 2x − 3 = 0: decider 16, a perfect square.x² − 2x − 3 = 0: decider 16, a perfect square.- Roots land exactly on 3 and −1.
x² − 4x − 1 = 0: decider 20, not a perfect square.- Roots sit between the integers:
2 + √5and2 − √5.
The rule
Take an equation with whole-number a, b and c. Its decider is above zero. A perfect square is a whole number times itself. For example, 16 = 4 × 4.
- A perfect square decider gives two rational roots. Each is a whole number or a fraction.
- Any other decider gives two irrational roots. Each has a square root in it, like
√5.
Irrational roots are still real. They are on the number line.
| Equation | Decider | Roots | Kind |
|---|---|---|---|
x² − 2x − 3 = 0 | 4 + 12 = 16 | 3 and −1 | rational |
x² − 4x − 1 = 0 | 16 + 4 = 20 | 2 + √5 and 2 − √5 | irrational |
2x² − 8x + 5 = 0 | 64 − 40 = 24 | (4 + √6)/2, (4 − √6)/2 | irrational |
For x² − 4x − 1 = 0, the formula gives (4 ± √20)/2. √20 is 2√5, because 20 = 4 × 5. So the roots are (4 ± 2√5)/2, which is 2 ± √5.
For 2x² − 8x + 5 = 0, the formula gives (8 ± √24)/4. √24 is 2√6, because 24 = 4 × 6. So the roots are (8 ± 2√6)/4, which is (4 ± √6)/2.
Keep the square root. A decimal is close, not exact.
Both kinds of root pass the same test. Put the root in, and the left side is 0.
Now try it. Pick an equation. Is its decider a perfect square?
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