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- Forming equations from roots
Forming equations from roots
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If you know the sum and product of two roots, you can write down the quadratic straight away.
The key result
A quadratic with roots p and q is
x2 − (sum of roots)x + (product of roots) = 0
Roots 4 and 7: sum 11, product 28, so x2 − 11x + 28 = 0.
x2 − (sum of roots)x + (product of roots) = 0
Roots 4 and 7: sum 11, product 28, so x2 − 11x + 28 = 0.
New roots from old
Start from α + β and αβ of the given equation. Work out the sum and the product of the new roots, then use the key result.
Example
α, β are roots of x2 − 4x + 1 = 0 (α + β = 4, αβ = 1).
New roots 2α and 2β:
sum = 2(α + β) = 8, product = 4αβ = 4
Equation: x2 − 8x + 4 = 0
New roots 2α and 2β:
sum = 2(α + β) = 8, product = 4αβ = 4
Equation: x2 − 8x + 4 = 0
α and β are the roots of 2x2 − 5x + 1 = 0. Find a quadratic equation with roots 1α and 1β.
- α + β = 52, αβ = 12
- Sum: (α + β)αβ = 5
- Product: 1αβ = 2
Answer: x2 − 5x + 2 = 0
x2 − (sum)x + (product) = 0. Find the new sum and product using α + β and αβ.
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