Choosing the method, checking the answer
You can solve by splitting or by the formula. The decider says which method works.
b² − 4ac, first. Step 1 of 4
- Read the decider,
b² − 4ac, first. - Perfect square, 0 too → factorise.
- Above zero, not a square → the formula.
- Below zero → no real answer.
The rule
Read the decider from the standard form before you choose. Take whole-number a, b and c.
- Perfect square, 0 too — split. The middle term splits, and the roots are rational.
- Above zero, not a square — use the formula.
- Below zero — say there is no real root. That is the answer.
The formula works for every equation with real roots. Splitting is quicker when the decider is a perfect square.
| Equation | Decider | Method | Answer |
|---|---|---|---|
2x² − 9x + 7 = 0 | 81 − 56 = 25 | split | 1 and 7/2 |
x² − 2x − 1 = 0 | 4 + 4 = 8 | formula | 1 + √2, 1 − √2 |
4x² − 12x + 9 = 0 | 144 − 144 = 0 | split | 3/2, counted twice |
x² + x + 3 = 0 | 1 − 12 = −11 | none | no real root |
To factorise is to write the left side as brackets multiplied. A split does that. 0 counts as a perfect square, because 0 = 0 × 0.
- Row 1: the pair −2 and −7 gives
(2x − 7)(x − 1) = 0. - Row 2: the formula gives
(2 ± √8)/2.√8is2√2, because 8 = 4 × 2. So the roots are1 ± √2. - Row 3: the pair −6 and −6 gives
(2x − 3)(2x − 3) = 0.
Check every root by putting it back. Take 7/2 in the first row. 2x² is 2 × 49/4 = 49/2, and 9x is 63/2. So 49/2 − 63/2 + 7 = −7 + 7 = 0.
Now try it. Work out b² − 4ac. Then pick a method.
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