Twice the same, or never
A decider can be exactly zero. It can also be below zero. Each case has its own verdict.
x² − 4x + 4 = 0 splits as (x − 2)(x − 2).x² − 4x + 4 = 0splits as(x − 2)(x − 2).- Both brackets are 0 at the same value:
x = 2. x² + x + 1 = 0: tryx = −1, 0, 1.- Values 1, 1, 3. None is zero. The decider, −3, says no value ever is.
The rule
When the decider is 0, the formula gives (−b + 0)/2a and (−b − 0)/2a. These are the same number. This is a repeated root: one value, counted twice.
Below zero, the formula needs the square root of a negative number. No real number squared gives a negative number. Say no real root.
Draw the left side as a curve. A real root is where the curve meets the x-axis. At a repeated root, the curve only touches the axis. With no real root, the curve stays off the axis.
| Decider | Equation | What you find |
|---|---|---|
| above zero | x² − 2x − 3 = 0 | 3 and −1, two values |
| zero | x² − 4x + 4 = 0 | 2, counted twice |
| below zero | x² + x + 1 = 0 | none, no real number |
Every quadratic equation fits exactly one row.
Find k for equal roots
Take x² − 4x + k = 0. Here a = 1, b = −4 and c = k. Equal roots need decider 0. The decider is (−4)² − 4 × 1 × k = 16 − 4k. Set it to 0: 16 − 4k = 0, so 4k = 16 and k = 4. The repeated root is −b/2a = 4/2 = 2.
Now try it. Slide c in x² − 4x + c = 0. Watch the curve meet the axis.
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