KCSE Quadratic Equations: How to Solve Them (3 Methods With Worked Examples)
Quadratic equations are one of the most frequently tested topics in KCSE Mathematics. They appear in Paper 1 (algebra section) and also show up in Paper 2 through topics like quadratic graphs, inequalities, and word problems. If you can solve quadratics confidently, you are securing marks across multiple sections of both papers.
What Is a Quadratic Equation?
A quadratic equation is any equation that can be written in the standard form:
ax^2 + bx + c = 0
where a, b, and c are constants and a is not equal to zero.
Examples:
- x^2 + 5x + 6 = 0 (a=1, b=5, c=6)
- 2x^2 - 3x - 2 = 0 (a=2, b=-3, c=-2)
- x^2 - 9 = 0 (a=1, b=0, c=-9)
The highest power of x is 2, which is what makes it "quadratic." You will always get two solutions (called roots), though sometimes both roots are the same number.
Method 1: Factorisation
Factorisation is the fastest method and the one KNEC expects you to try first when the equation factorises neatly.
How It Works
You rewrite the quadratic expression as a product of two brackets, then use the fact that if AB = 0, then either A = 0 or B = 0.
Worked Example 1
Solve x^2 + 5x + 6 = 0
Step 1: Find two numbers that multiply to give +6 and add to give +5.
- 2 and 3 work: 2 x 3 = 6, and 2 + 3 = 5.
Step 2: Write the factorised form.
- (x + 2)(x + 3) = 0
Step 3: Set each bracket equal to zero.
- x + 2 = 0, so x = -2
- x + 3 = 0, so x = -3
Answer: x = -2 or x = -3
Worked Example 2
Solve 2x^2 - 3x - 2 = 0
Step 1: Multiply a and c: 2 x (-2) = -4. Find two numbers that multiply to give -4 and add to give -3.
- -4 and +1 work: -4 x 1 = -4, and -4 + 1 = -3.
Step 2: Split the middle term.
- 2x^2 - 4x + x - 2 = 0
Step 3: Factorise by grouping.
- 2x(x - 2) + 1(x - 2) = 0
- (2x + 1)(x - 2) = 0
Step 4: Solve.
- 2x + 1 = 0, so x = -1/2
- x - 2 = 0, so x = 2
Answer: x = -1/2 or x = 2
When to Use Factorisation
Use factorisation when the equation has integer or simple fractional roots. If you cannot find factors after 30 seconds, move to another method.
Method 2: Completing the Square
Completing the square is often required in KCSE when the question specifically asks for it, or when you need to find the turning point of a quadratic graph.
How It Works
You rewrite the equation in the form (x + p)^2 = q, then solve by taking the square root of both sides.
Worked Example
Solve x^2 + 6x + 2 = 0 by completing the square
Step 1: Move the constant to the right side.
- x^2 + 6x = -2
Step 2: Take half the coefficient of x, square it, and add to both sides. Half of 6 is 3, and 3^2 = 9.
- x^2 + 6x + 9 = -2 + 9
- x^2 + 6x + 9 = 7
Step 3: Write the left side as a perfect square.
- (x + 3)^2 = 7
Step 4: Take the square root of both sides.
- x + 3 = +/-sqrt(7)
- x = -3 + sqrt(7) or x = -3 - sqrt(7)
Step 5: Calculate (if the question asks for decimal answers).
- sqrt(7) = 2.646
- x = -3 + 2.646 = -0.354 or x = -3 - 2.646 = -5.646
Answer: x = -0.354 or x = -5.646 (to 3 decimal places)
Important Note for KCSE
When the coefficient of x^2 is not 1, divide the entire equation by that coefficient first. For example, to complete the square on 2x^2 + 8x + 3 = 0, start by dividing everything by 2 to get x^2 + 4x + 3/2 = 0.
Method 3: The Quadratic Formula
The quadratic formula works for every quadratic equation, including those that do not factorise neatly. It is the method of last resort and the most reliable.
x = (-b +/- sqrt(b^2 - 4ac)) / 2a
Worked Example
Solve 3x^2 + 2x - 4 = 0, giving your answers correct to 3 decimal places
Step 1: Identify a, b, and c.
- a = 3, b = 2, c = -4
Step 2: Calculate the discriminant (b^2 - 4ac).
- b^2 - 4ac = (2)^2 - 4(3)(-4) = 4 + 48 = 52
Step 3: Apply the formula.
- x = (-2 +/- sqrt(52)) / (2 x 3)
- x = (-2 +/- 7.211) / 6
Step 4: Calculate both roots.
- x = (-2 + 7.211) / 6 = 5.211 / 6 = 0.869
- x = (-2 - 7.211) / 6 = -9.211 / 6 = -1.535
Answer: x = 0.869 or x = -1.535
The Discriminant Tells You What to Expect
The value b^2 - 4ac (called the discriminant) reveals the nature of the roots before you solve:
| Discriminant | Nature of Roots | |-------------|-----------------| | b^2 - 4ac > 0 | Two distinct real roots | | b^2 - 4ac = 0 | Two equal real roots (repeated root) | | b^2 - 4ac < 0 | No real roots |
KCSE sometimes asks you to "determine the nature of the roots" without actually solving the equation. Calculate the discriminant and use the table above.
Word Problems That Produce Quadratics
KCSE frequently tests quadratic equations through word problems. The key skill is translating the words into an equation.
Common Setup
"The length of a rectangular garden is 3 metres more than its width. The area of the garden is 40 m^2. Find the dimensions of the garden."
Solution:
Let the width = x metres. Then the length = (x + 3) metres.
Area = length x width:
- x(x + 3) = 40
- x^2 + 3x - 40 = 0
Factorise: (x + 8)(x - 5) = 0
So x = -8 or x = 5. Since width cannot be negative, x = 5.
Answer: Width = 5 m, Length = 8 m
Always check whether both roots make sense in the context of the problem. Lengths, ages, and quantities cannot be negative.
Common Mistakes to Avoid
- Forgetting to set the equation to zero. You cannot factorise x^2 + 5x = 6 directly. Move the 6 first: x^2 + 5x - 6 = 0.
- Sign errors when factorising. If c is negative, one of your factor numbers must be negative. Double-check by expanding your brackets.
- Dropping the negative root from the formula. The "+/-" in the quadratic formula means you must calculate two values. Do not stop at one.
- Not simplifying. If the question asks for exact answers, leave your answer in surd form. If it asks for decimal places, round correctly.
- Dividing by x. Never divide both sides by x to "simplify" the equation. You will lose a root (x = 0 is a valid solution if it satisfies the equation).
How Quadratic Equations Appear in KCSE
Based on past papers, expect quadratic equations in these forms:
- Direct "solve" questions (4-6 marks)
- Word problems requiring you to form and solve a quadratic (6-8 marks)
- Questions on quadratic graphs asking for x-intercepts
- Completing the square to find the turning point of a parabola
- Nature of roots using the discriminant
Mastering all three methods means you are prepared regardless of how the question is framed.
Start Practising Now
The best way to get faster at quadratics is to practise a high volume of questions. Work through KCSE-style problems on HighMarks:
- Quadratic Equations and Inequalities -- topic-specific practice
- Algebraic Expressions -- build your factorisation skills
- Equations of Straight Lines -- related coordinate geometry
- All Mathematics topics -- full topic list for Paper 1 and Paper 2
Start practising quadratic equations on HighMarks and build the speed and accuracy you need for exam day.