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Quadratic Functions and Equations in One Variable · Form 4

Quadratic Functions and Equations in One Variable: Worked Examples (Medium)

This set moves to the quadratic formula, forming an equation from its roots, and sketching a parabola, problems that take two or three linked steps. For students ready to combine skills.

Written by the spmmath.com.my editorial team.

Worked example 1

Solve 2x² + 3x − 4 = 0 using the quadratic formula, giving your answers correct to two decimal places.

  1. Identify a = 2, b = 3, c = −4.
  2. Find the discriminant: b² − 4ac = 3² − 4(2)(−4) = 9 + 32 = 41.
  3. Substitute into x = (−b ± √(b² − 4ac)) / (2a) = (−3 ± √41) / 4.
  4. Since √41 ≈ 6.403, x = (−3 + 6.403)/4 or x = (−3 − 6.403)/4.
  5. Work out both: x = 3.403/4 = 0.85 or x = −9.403/4 = −2.35.

Worked example 2

A quadratic equation has roots x = 2 and x = −5. Form the equation in the form x² + bx + c = 0.

  1. If the roots are 2 and −5, the factors are (x − 2) and (x + 5).
  2. Write the equation as (x − 2)(x + 5) = 0.
  3. Expand: x² + 5x − 2x − 10 = 0.
  4. Simplify the middle terms: x² + 3x − 10 = 0.

Worked example 3

For the curve y = x² − 2x − 3, find the x-intercepts, the y-intercept and the turning point, then describe the shape of the graph.

  1. x-intercepts: set y = 0 and factorise x² − 2x − 3 = (x − 3)(x + 1) = 0, so x = 3 or x = −1.
  2. y-intercept: set x = 0, giving y = −3, so the curve crosses the y-axis at (0, −3).
  3. Turning point by completing the square: y = (x − 1)² − 1 − 3 = (x − 1)² − 4.
  4. The turning point is (1, −4), and since the coefficient of x² is positive it is a minimum.
  5. The graph is a parabola opening upward, lowest at (1, −4), crossing the x-axis at (−1, 0) and (3, 0).

Worked example 4

Solve 3x² − 7x + 2 = 0 by factorising.

  1. Multiply the coefficient of x² by the constant: 3 × 2 = 6.
  2. Find two numbers whose product is 6 and whose sum is −7: they are −6 and −1.
  3. Split the middle term: 3x² − 6x − x + 2 = 0.
  4. Group and factorise: 3x(x − 2) − 1(x − 2) = 0, so (3x − 1)(x − 2) = 0.
  5. Set each factor to zero: 3x − 1 = 0 gives x = 1/3, or x − 2 = 0 gives x = 2.

Worked example 5

Express y = x² + 6x + 5 in the form (x + p)² + q by completing the square, and state the minimum point of the graph.

  1. Halve the coefficient of x: 6 ÷ 2 = 3.
  2. Write x² + 6x = (x + 3)² − 3² = (x + 3)² − 9.
  3. Substitute back: y = (x + 3)² − 9 + 5 = (x + 3)² − 4.
  4. The graph is lowest when (x + 3)² = 0, i.e. x = −3, giving y = −4.

Worked example 6

The quadratic equation x² + kx + 9 = 0 has two equal roots. Find the possible values of k.

  1. Two equal roots means the discriminant b² − 4ac = 0.
  2. Here a = 1, b = k and c = 9.
  3. Substitute: k² − 4(1)(9) = 0, so k² − 36 = 0.
  4. Solve: k² = 36, so k = 6 or k = −6.

Worked example 7

Solve 2x² − 4x − 3 = 0 using the quadratic formula, giving your answers correct to two decimal places.

  1. Identify a = 2, b = −4, c = −3.
  2. Substitute into x = [−b ± √(b² − 4ac)] / (2a): x = [4 ± √(16 + 24)] / 4 = [4 ± √40] / 4
  3. √40 = 6.3246…
  4. x = (4 + 6.3246)/4 = 2.58 or x = (4 − 6.3246)/4 = −0.58

Worked example 8

The quadratic equation 2x² − 4x + k = 0 has two equal roots. Find the value of k.

  1. For equal roots, the discriminant b² − 4ac = 0.
  2. Here a = 2, b = −4, c = k: (−4)² − 4(2)(k) = 0
  3. 16 − 8k = 0
  4. k = 2

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)

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Frequently asked questions

When should I use completing the square instead of the quadratic formula?

Completing the square is essential when the question asks for the turning point, maximum/minimum value, or axis of symmetry, the formula alone doesn't reveal these directly. Use the quadratic formula mainly when you just need the roots quickly and factorisation isn't obvious.

What's a common error when applying the quadratic formula?

Students often forget the ± sign, mis-substitute a negative b, or make an arithmetic slip inside the square root (b² − 4ac). Substitute carefully, keep track of signs using brackets, and simplify the discriminant fully before taking the square root.

How does the discriminant help without solving the whole equation?

The discriminant b² − 4ac tells you the nature of the roots before you solve: positive means two different real roots, zero means one repeated root, negative means no real roots. Examiners often ask you to state this without actually finding the roots themselves.

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