Roots and coefficients (NF)
A standard method from Quadratic equations in one unknown: the steps and worked examples.
Steps
For ax² + bx + c = 0 with roots α, β:
- α + β = −b/a, αβ = c/a.
- Useful identities: α² + β² = (α + β)² − 2αβ; 1/α + 1/β = (α + β)/αβ.
- An equation with roots p and q: x² − (p + q)x + pq = 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.