α and β are the roots of 2x2 − 6x + 1 = 0. Find α2 + β2.
- a = 2, b = −6, c = 1, so α + β = −(−6)/2 = 3 and αβ = 1/2.
- α2 + β2 = (α + β)2 − 2αβ = 32 − 2 × (1/2).
- = 9 − 1 = 8.
Answer: α2 + β2 = 8
Roots, coefficients and nature of roots
Maths · Quadratic equations. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
For the quadratic equation ax2 + bx + c = 0 (a ≠ 0) with roots α and β, you can find the sum of the roots and the product of the roots straight from the coefficients, without solving. Any symmetric expression, such as α2 + β2 or 1/α + 1/β, can then be written using only the sum and the product.
The same two numbers let you form an equation from its roots. The discriminant b2 − 4ac tells you the nature of the roots: real or complex, equal or unequal, rational or irrational.
For one linear and one quadratic equation together, make one variable the subject of the linear equation and substitute it into the other.
Rules to remember
- sum of roots = −(coefficient of x) ÷ (coefficient of x2): α + β = −b/a
- product of roots = constant term ÷ (coefficient of x2): αβ = c/a
- Equation with given roots: x2 − (sum of roots)x + (product of roots) = 0
- α2 + β2 = (α + β)2 − 2αβ and 1/α + 1/β = (α + β)/(αβ)
- b2 − 4ac > 0: roots real and unequal. They are rational if b2 − 4ac is a perfect square and irrational if it is not (a, b, c rational)
- b2 − 4ac = 0: roots real (rational) and equal, each −b/(2a)
- b2 − 4ac < 0: roots complex (imaginary) and unequal
Common mistake
Students forget the minus sign in the sum of the roots. The sum is −b/a, not b/a, so for x2 − 5x + 4 = 0 the sum is +5.
Worked example
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