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Quadratic expressions and equations — KCSE Mathematics

KCSE Mathematics · 105 practice questions · 4 syllabus objectives · 4 revision lessons

36 easy42 medium27 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Factorise quadratic expressions of the form ax² + bx + c; solve quadratic equations by factorisation

Apply the quadratic formula x = (–b ± √(b² – 4ac)) / 2a to solve quadratic equations and determine the nature of roots using the discriminant

Solve quadratic equations by completing the square; form a quadratic equation from given roots

Quadratic expressions and equations

Revision Notes

Concise lesson notes for Quadratic expressions and equations, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Factorising Quadratic Expressions

To factorise a quadratic expression of the form ax² + bx + c, follow these steps:

  1. Identify coefficients: a (leading coefficient), b (linear coefficient), and c (constant).
  2. Look for two numbers that multiply to ac and add to b.
  3. Rewrite the middle term using these two numbers.
  4. Group the terms and factor out the common factors.
  5. Write the expression as a product of two binomials.

Example: Factorise the expression 2x² + 7x + 3.

  • Here, a = 2, b = 7, c = 3.
  • Multiply a and c: 2 * 3 = 6.
  • Find two numbers that multiply to 6 and add to 7: 6 and 1.
  • Rewrite: 2x² + 6x + 1x + 3.
  • Group: (2x² + 6x) + (1x + 3).
  • Factor: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

To solve quadratic equations by factorisation, set the expression to zero and solve for x by setting each factor to zero.

Key points to remember

  • Identify coefficients a, b, and c in the quadratic expression.
  • Find two numbers that multiply to ac and add to b.
  • Rewrite the quadratic expression using these two numbers.
  • Group terms and factor out common factors.
  • Set each factor to zero to find x.

Worked example

Factorise and solve: x² + 5x + 6 = 0.

  • Factor: (x + 2)(x + 3) = 0.
  • Set factors to zero: x + 2 = 0 or x + 3 = 0.
  • Solutions: x = -2 or x = -3.

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More lessons in this topic

Lesson 2: Solving Quadratic Equations with the Formula

Objective: Apply the quadratic formula x = (–b ± √(b² – 4ac)) / 2a to solve quadratic equations and determine the nature of roots using the discriminant

To solve a quadratic equation in the form ax² + bx + c = 0, we can apply the quadratic formula:

x = (–b ± √(b² – 4ac)) / 2a.

The term (b² – 4ac) is known as the discriminant and helps us determine the nature of the roots:

  • If D > 0, there are two distinct real roots.
  • If D = 0, there is one real root (a repeated root).
  • If D < 0, the roots are complex (not real).

Example: Solve the equation 2x² - 4x + 2 = 0.

  1. Identify coefficients: a = 2, b = -4, c = 2.
  2. Calculate the discriminant: D = (-4)² - 4(2)(2) = 16 - 16 = 0.
  3. Since D = 0, there is one real root.
  4. Use the quadratic formula:
    x = (–(-4) ± √0) / (2*2) = 4 / 4 = 1.
    Thus, the equation has one real root: x = 1.
  • Use the quadratic formula to find roots of quadratic equations.
  • Calculate the discriminant to determine root nature.
  • Identify roots as real or complex based on discriminant value.

Solve: x² - 3x + 2 = 0.

  • Coefficients: a = 1, b = -3, c = 2.
  • D = (-3)² - 4(1)(2) = 9 - 8 = 1 (D > 0).
  • Roots are x = (3 ± √1) / 2 = (3 ± 1) / 2.
  • Roots are x = 2 and x = 1.
Lesson 3: Solving Quadratic Equations by Completing the Square

Objective: Solve quadratic equations by completing the square; form a quadratic equation from given roots

To solve quadratic equations by completing the square, follow these steps:

  1. Start with the standard form of the quadratic equation: ax² + bx + c = 0.
  2. Divide all terms by 'a' if 'a' is not equal to 1.
  3. Move the constant term to the other side of the equation.
  4. Take half of the coefficient of 'x', square it, and add it to both sides.
  5. Factor the left-hand side into a perfect square trinomial.
  6. Solve for 'x' by taking the square root of both sides and isolating 'x'.

Example: Solve the equation x² - 6x + 5 = 0 by completing the square.

  • Move 5 to the right: x² - 6x = -5.
  • Take half of -6, square it: (-3)² = 9.
  • Add 9 to both sides: x² - 6x + 9 = 4.
  • Factor: (x - 3)² = 4.
  • Take the square root: x - 3 = ±2.
  • Solve for x: x = 5 or x = 1.

To form a quadratic equation from given roots, use the formula: (x - r₁)(x - r₂) = 0, where r₁ and r₂ are the roots.

  • Complete the square to rewrite the quadratic equation.
  • Factor the perfect square trinomial formed.
  • Use roots to form a quadratic equation.

Given roots 2 and 3, form the quadratic equation:

  • (x - 2)(x - 3) = 0.
  • Expand: x² - 5x + 6 = 0.
Lesson 4: Understanding Quadratic Expressions and Equations

Objective: Quadratic expressions and equations

A quadratic expression is a polynomial of the form ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic expression is a parabola. Quadratic equations are formed when a quadratic expression is set equal to zero, represented as ax² + bx + c = 0.

To solve quadratic equations, we can use:

  • Factoring: Rewrite the equation in factored form.
  • Completing the square: Rearranging the equation to form a perfect square.
  • Quadratic formula: Use x = (-b ± √(b² - 4ac)) / (2a).

For example, consider the quadratic equation x² - 5x + 6 = 0.

  • Factoring: (x - 2)(x - 3) = 0.
  • Solutions: x = 2 and x = 3.

Remember, the discriminant (b² - 4ac) helps determine the nature of the roots:

  • If positive, two distinct real roots.
  • If zero, one real root.
  • If negative, no real roots.
  • Quadratic expressions are in the form ax² + bx + c.
  • Quadratic equations are set to zero: ax² + bx + c = 0.
  • Roots can be found using factoring, completing the square, or the quadratic formula.
  • The discriminant indicates the nature of the roots.

Solve the quadratic equation x² - 4x - 5 = 0.

  • Factor: (x - 5)(x + 1) = 0.
  • Roots: x = 5 and x = -1.

Sample Questions

Read 3 questions and answers free. Sign up to access all 105 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER3 marks

State how to solve the quadratic equation 2x² - 8x + 6 = 0 by completing the square. (3 marks)

Answer & marking scheme

Part (a) — 3 marks
Divide the entire equation by 2 to simplify it (1 mk)
Rearrange to form (x - 4)² = 4 (1 mk)
Solve for x by taking the square root and isolating x (1 mk)
2
easySHORT ANSWER2 marks

Given the quadratic equation x² + 6x + 9 = 0, use the discriminant to determine the nature of its roots. (2 marks)

Answer & marking scheme

Part (a) — 1 mark
Correctly identified and calculated the discriminant (b² - 4ac = 0) (1 mk)
Part (b) — 1 mark
The roots are real and equal (1 mk)
3
easySHORT ANSWER4 marks

Solve the quadratic equation 2x² - 4x - 6 = 0 using the quadratic formula and state the nature of its roots. (4 marks)

Answer & marking scheme

Part (a) — 3 marks
Substituted values into the quadratic formula correctly (1 mk)
Calculated the discriminant (b² - 4ac) correctly (1 mk)
Obtained the correct roots (x = 3 or x = -1) from the formula (1 mk)
Part (b) — 1 mark
The roots are real and distinct (1 mk)
4

Explain how to factorise the quadratic equation x² - 5x + 6 = 0. (4 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Quadratic expressions and equations" cover?

Quadratic expressions and equations covers Factorise quadratic expressions of the form ax² + bx + c; solve quadratic equations by factorisation; Apply the quadratic formula x = (–b ± √(b² – 4ac)) / 2a to solve quadratic equations and determine the nature of roots using the discriminant; Solve quadratic equations by completing the square; form a quadratic equation from given roots, and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Quadratic expressions and equations?

HighMarks has 105 Quadratic expressions and equations practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Quadratic expressions and equations for the KCSE exam?

Start with the revision notes on this page to refresh the core concepts, then work through the practice questions in increasing difficulty. Sign up for HighMarks to get a personalised study plan that adapts to the topics you keep getting wrong, plus mock exams, subject-wide practice, and detailed performance tracking. See pricing.

Why Practise Quadratic expressions and equations?

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