Further Mathematics · Lesson 6

Quadratic functions: roots, discriminant, sum & product of roots

WAECNECO

What the syllabus asks

WAEC: Pure Mathematics: quadratic functions and equations — roots, discriminant, nature of roots, sum and product of roots, forming equations, maximum and minimum, quadratic inequalities

NECO: Pure Mathematics: quadratic functions and equations — roots, discriminant, nature of roots, sum and product of roots, forming equations, maximum and minimum, quadratic inequalities

What you will learn

Solve quadratics by factorisation, completing the square and the formula.
Use the discriminant to state the nature of roots and find unknown constants.
Find the sum and product of roots and evaluate symmetric expressions.
Form quadratic equations with given or transformed roots.
Find maximum or minimum values, the axis of symmetry and the range.
Solve quadratic inequalities and conditions for a quadratic to be always positive.

Key points

f(x) = ax² + bx + c: ∪ if a > 0 (minimum), ∩ if a < 0 (maximum); y-intercept c; axis x = −b/2a.
Solve by factorising, completing the square, or x = [−b ± √(b² − 4ac)]/2a.
D = b² − 4ac: > 0 distinct real; = 0 equal; < 0 no real roots; "real" means D ≥ 0.
α + β = −b/a; αβ = c/a.
α² + β² = (α+β)² − 2αβ; (α−β)² = (α+β)² − 4αβ; α³ + β³ = (α+β)³ − 3αβ(α+β).
Equation with roots of sum S and product P: x² − Sx + P = 0.
a(x − h)² + k gives vertex (h, k) and range.
Inequalities: find roots, sketch; always positive needs a > 0 and D < 0; tangent needs D = 0.

Start here: The quadratic function and its graph

A quadratic function is f(x) = ax² + bx + c, where a, b, c are constants and a ≠ 0. If a were 0, the x² term would disappear and the function would be linear.

The graph of every quadratic is a symmetrical curve called a parabola. When a ball is thrown at an inter-house sports meeting, its path is (very nearly) a parabola; so is the shape of water from a hose. The sign of a decides the shape:

a > 0: the curve opens upward (∪) and has a minimum point.
a < 0: the curve opens downward (∩) and has a maximum point.

The graph cuts the y-axis where x = 0, so the y-intercept is c. It cuts the x-axis at the roots of ax² + bx + c = 0, if there are any. The curve is symmetrical about a vertical line, the axis of symmetry, which passes through the turning point.

For f(x) = ax² + bx + c: y-intercept = c; axis of symmetry x = −b/(2a); the vertex lies on this axis.

Worked example 6.1

For f(x) = x² − 4x − 12, find the y-intercept, the axis of symmetry and the x-intercepts.

y-intercept: f(0) = −12.
Axis: x = −(−4)/(2 × 1) = 2.
x-intercepts: x² − 4x − 12 = (x − 6)(x + 2) = 0, so x = 6 or x = −2.
Check: the axis x = 2 is midway between −2 and 6 ✓. We check this because the roots of a parabola are always symmetrical about its axis.

(0, −12); x = 2; (−2, 0) and (6, 0)

⚠️ Common mistake: writing the axis as x = b/2a, forgetting the minus sign.

Try it yourself. State whether f(x) = −3x² + 2x + 7 has a maximum or minimum, and find its axis of symmetry.

Show solution

a = −3 < 0: maximum. Axis x = −2/(2 × −3) = 1/3.

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Practice questions

1. Solve 3x² + 5x − 2 = 0.

Show worked answer

(3x − 1)(x + 2) = 0: x = 1/3 or −2.

2. State the nature of the roots of 4x² − 4x + 1 = 0.

Show worked answer

D = 16 − 16 = 0: equal roots (x = ½).

3. Find k if x² − 2kx + 2k + 3 = 0 has equal roots.

Show worked answer

4k² − 8k − 12 = 0 → k² − 2k − 3 = 0 → k = 3 or −1.

4. If α, β are roots of x² − 5x + 3 = 0, find α² + β².

Show worked answer

25 − 6 = 19.

5. Form the equation whose roots are −3 and 1/2.

Show worked answer

S = −5/2, P = −3/2: 2x² + 5x − 3 = 0.

Key words

Quadratic function: A function of the form ax² + bx + c with a ≠ 0.
Root (zero): A value of x for which f(x) = 0.
Discriminant: D = b² − 4ac.
Parabola: The curve that is the graph of a quadratic.
Vertex (turning point): The highest or lowest point of a parabola.
Axis of symmetry: The vertical line x = −b/2a through the vertex.
Completing the square: Rewriting ax² + bx + c as a(x − h)² + k.
Real roots: Roots that are real numbers.
Equal (repeated) roots: Both roots the same, when D = 0.
Symmetric function of roots: An expression unchanged when α and β are swapped, e.g. α² + β².
Quadratic inequality: An inequality such as ax² + bx + c > 0.
Range: The set of output values of a function.

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