Nature of roots
A standard method from Quadratic equations in one unknown: the steps and worked examples.
Steps
| Δ = b² − 4ac | Roots | Graph of y = ax² + bx + c |
|---|---|---|
| Δ > 0 | two distinct real roots | cuts the x-axis twice |
| Δ = 0 | two equal real roots | touches the x-axis once |
| Δ < 0 | no real roots | doesn't meet the x-axis |
"Has real roots" means Δ ≥ 0.
Want to check an answer? Quadratic equation calculator shows every step.
Worked examples
Example 1: x² + (k − 2)x + 4 = 0 has equal roots. Find k.
Δ = (k − 2)² − 4(1)(4) = 0
(k − 2)² = 16
k − 2 = ±4
k = 6 or k = −2Example 2: NF: α and β are the roots of x² − 3x − 5 = 0. Find α² + β² and form an equation with roots α + 1 and β + 1.
α + β = 3, αβ = −5
α² + β² = 3² − 2(−5) = 19
sum = (α + 1) + (β + 1) = 5
product = αβ + (α + β) + 1 = −5 + 3 + 1 = −1
equation: x² − 5x − 1 = 0For the whole topic, read Quadratic equations in one unknown: key points, common mistakes and how the DSE tests it.