If α and β are the roots of x2 − 4x + 1 = 0, then α3 + β3 is
- Sum: α + β = −(−4)/1 = 4. Product: αβ = 1/1 = 1.
- α3 + β3 = (α + β)3 − 3αβ(α + β) = 43 − 3 × 1 × 4.
- 64 − 12 = 52.
Answer: 52
Quadratic equations
NUST NET (Engineering) · Maths · Quadratic equations. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
The idea
A quadratic equation ax2 + bx + c = 0 (a ≠ 0) has two roots. Factorise if the factors are easy to see; otherwise use the formula. Some harder equations become quadratic after a substitution, such as y = x2 or y = √x. Solve for y, then go back to x and reject any value that is not allowed.
The discriminant b2 − 4ac tells you the nature of the roots without solving. The sum and product of roots come straight from the coefficients, so expressions in α and β can be found without knowing α and β themselves.
The cube roots of unity are 1, ω and ω2. Two facts about them answer almost every question.
Rules to remember
- x = (−b ± √(b2 − 4ac))/(2a)
- b2 − 4ac > 0: real and distinct roots; = 0: real and equal; < 0: complex conjugate roots
- With rational a, b, c: the roots are rational when b2 − 4ac is a perfect square, irrational when it is positive but not a perfect square
- Sum of roots α + β = −b/a; product αβ = c/a
- Equation with given roots: x2 − (sum of roots)x + (product of roots) = 0
- α2 + β2 = (α + β)2 − 2αβ; (α − β)2 = (α + β)2 − 4αβ; α3 + β3 = (α + β)3 − 3αβ(α + β)
- Cube roots of unity: 1 + ω + ω2 = 0 and ω3 = 1, so reduce any power of ω using its remainder on division by 3
Common mistake
Dropping the minus sign in the sum of roots. The sum is −b/a, not b/a: for x2 − 5x + 3 = 0 the sum is +5, and for x2 + 5x + 3 = 0 it is −5.
Worked example
Practice questions
Try each one, then open the answer.
1. The solutions of x1/2 − 5x1/4 + 6 = 0 are
- A4 and 9
- B2 and 3
- C16 and 81
- D8 and 27
Show answer
Answer: C. Let y = x1/4; then y2 − 5y + 6 = 0 gives y = 2 or 3. So x = 24 = 16 or 34 = 81; 2 and 3 are the values of y, not x.
2. The roots of x2 − (√3 + 1)x + √3 = 0 are
- A1 and √3
- B−1 and −√3
- C1 and −√3
- D−1 and √3
Show answer
Answer: A. Two numbers with sum √3 + 1 and product √3 are 1 and √3, so the equation is (x − 1)(x − √3) = 0. Negative roots would make the sum negative.
3. The roots of 2x2 − 4x + 3 = 0 are
- Areal and equal
- Breal, rational and unequal
- Creal, irrational and unequal
- Dimaginary (complex conjugates)
Show answer
Answer: D. The discriminant b2 − 4ac = 16 − 24 = −8 is negative, so the roots are non-real and form a conjugate pair.
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