Quadratic equations are one of the most frequently tested topics in ZIMSEC O-Level Mathematics (4004/4008). This guide walks through every method with fully worked examples.
What is a Quadratic Equation?
A quadratic equation is any equation of the form:
[ ax^2 + bx + c = 0 ]
where ( a eq 0 ) and ( a, b, c ) are real numbers. The solution(s) are called roots.
Method 1 — Factorisation
Use when the equation factors neatly into integers. Works quickly for “nice” numbers.
Example: Solve ( x^2 - 5x + 6 = 0 )
- Find two numbers that multiply to ( +6 ) and add to ( -5 ): they are ( -2 ) and ( -3 ).
- Write: ( (x - 2)(x - 3) = 0 )
- Set each factor to zero: ( x = 2 ) or ( x = 3 )
Check: ( 4 - 10 + 6 = 0 checkmark ) and ( 9 - 15 + 6 = 0 checkmark )
Method 2 — Completing the Square
Use when factorisation fails or the question specifically asks for it.
Example: Solve ( x^2 + 4x - 1 = 0 )
- Move the constant: ( x^2 + 4x = 1 )
- Add ( left(dfrac{4}{2} ight)^2 = 4 ) to both sides: ( x^2 + 4x + 4 = 5 )
- Write as perfect square: ( (x + 2)^2 = 5 )
- Take square root: ( x + 2 = pmsqrt{5} )
- Solve: ( x = -2 + sqrt{5} approx 0.24 ) or ( x = -2 - sqrt{5} approx -4.24 )
Method 3 — The Quadratic Formula
Works for any quadratic. Always reliable in exams when other methods fail.
[ x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ]
Example: Solve ( 2x^2 - 3x - 2 = 0 ) Here ( a=2,; b=-3,; c=-2 )
- Discriminant: ( Delta = (-3)^2 - 4(2)(-2) = 9 + 16 = 25 )
- ( x = dfrac{3 pm sqrt{25}}{4} = dfrac{3 pm 5}{4} )
- ( x = dfrac{8}{4} = 2 ) or ( x = dfrac{-2}{4} = -dfrac{1}{2} )
The Discriminant ( Delta = b^2 - 4ac )
| Value of (Delta) | Nature of Roots |
|---|---|
| (Delta > 0) | Two distinct real roots |
| (Delta = 0) | One repeated real root (equal roots) |
| (Delta < 0) | No real roots (complex roots) |
ZIMSEC Exam-Style Practice Questions
- Solve ( x^2 - 7x + 12 = 0 ) by factorisation.
- Solve ( 3x^2 + 5x - 2 = 0 ) using the quadratic formula, giving answers to 2 decimal places.
- The area of a rectangle is ( 60 , ext{cm}^2 ). Its length is ( (x+4) ) cm and its width is ( x ) cm. Form a quadratic equation and find ( x ).
- Show that ( x^2 + x + 1 = 0 ) has no real roots.
Download the full solution sheet (PDF) below.
Comments (4)
This explanation is so much clearer than my textbook! The discriminant table especially helped me understand why some equations have no real roots.
Thank you Mrs Chikwanda! We will be adding more algebra topics next week including simultaneous equations.
Excellent resource. I share these with my Form 4 class at the beginning of each revision unit. The worked examples follow the same style ZIMSEC examiners use.
Can you please post the solutions to the practice questions at the bottom? I got question 3 right but not sure about question 2.
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