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Quadratic Functions and Equations in One Variable · Form 4

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

This set drills solving quadratic equations by factorising, the first method every Form 4 student should master. Best for building confidence before the harder techniques.

Written by the spmmath.com.my editorial team.

Worked example 1

Solve the quadratic equation x² + 7x + 12 = 0 by factorising.

  1. Look for two numbers that multiply to +12 and add to +7.
  2. The numbers are 3 and 4, so factorise as (x + 3)(x + 4) = 0.
  3. Set each factor to zero: x + 3 = 0 or x + 4 = 0.
  4. Solve each: x = −3 or x = −4.

Worked example 2

Solve x² − 5x − 14 = 0 by factorising.

  1. Find two numbers that multiply to −14 and add to −5.
  2. The numbers are −7 and +2, so factorise as (x − 7)(x + 2) = 0.
  3. Set each factor to zero: x − 7 = 0 or x + 2 = 0.
  4. Solve each: x = 7 or x = −2.

Worked example 3

Solve x² − 49 = 0 by factorising.

  1. Recognise a difference of two squares: 49 = 7², so x² − 49 = x² − 7².
  2. Factorise using a² − b² = (a − b)(a + b): (x − 7)(x + 7) = 0.
  3. Set each factor to zero: x − 7 = 0 or x + 7 = 0.
  4. Solve each: x = 7 or x = −7.

Worked example 4

Solve the quadratic equation x² − 8x + 15 = 0 by factorising.

  1. Look for two numbers whose product is +15 and whose sum is −8.
  2. Both numbers are negative: −3 and −5, since (−3)(−5) = 15 and (−3) + (−5) = −8.
  3. Factorise as (x − 3)(x − 5) = 0.
  4. Set each factor to zero: x − 3 = 0 or x − 5 = 0, so x = 3 or x = 5.

Worked example 5

Solve x² + 3x − 10 = 0 by factorising.

  1. Find two numbers whose product is −10 and whose sum is +3.
  2. The numbers are +5 and −2, since 5 × (−2) = −10 and 5 + (−2) = 3.
  3. Factorise as (x + 5)(x − 2) = 0.
  4. Set each factor to zero: x + 5 = 0 or x − 2 = 0, so x = −5 or x = 2.

Worked example 6

Solve x² − 6x = 0 by factorising.

  1. There is no constant term, so take out the common factor x.
  2. Factorise as x(x − 6) = 0.
  3. Set each factor to zero: x = 0 or x − 6 = 0.
  4. So x = 0 or x = 6.

Worked example 7

Solve x² − 9x + 20 = 0 by factorising.

  1. Find two numbers that multiply to give 20 and add to give −9: −4 and −5.
  2. Factorise: (x − 4)(x − 5) = 0
  3. x − 4 = 0 or x − 5 = 0
  4. x = 4 or x = 5

Worked example 8

Solve x² + 5x − 24 = 0 by factorising.

  1. Find two numbers that multiply to give −24 and add to give 5: 8 and −3.
  2. Factorise: (x + 8)(x − 3) = 0
  3. x + 8 = 0 or x − 3 = 0
  4. x = −8 or x = 3

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

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Frequently asked questions

What's the easiest way to check if a function is quadratic?

Look for the highest power of x, if it's x² (and no higher power), in the general form f(x) = ax² + bx + c with a ≠ 0, it's quadratic. Many students lose marks by allowing a = 0, which actually makes it a linear function instead.

When solving quadratic equations by factorisation, what mistake costs the most marks?

Forgetting to rearrange the equation to ax² + bx + c = 0 before factorising, or dropping a negative sign when splitting the middle term. Always set it to zero first, factorise fully into two brackets, then set each bracket equal to zero separately to get both roots.

Why do I need to sketch quadratic graphs, and what should the sketch show?

Sketches test whether you understand the shape, not just the algebra. A correct sketch shows the opening direction (from the sign of a), the x-intercepts (roots), and the turning point or axis of symmetry, examiners give marks for each labelled feature, not just a rough curve.

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