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CMA Foundation · Fundamentals of Business Mathematics and Statistics

Quadratic Equations for CMA Foundation: Chapter Guide

A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. You solve it by factorisation or by the formula x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac tells you the nature of the roots. The sum of roots is −b/a and the product is c/a.

What this chapter covers

This chapter deals with equations where the highest power of the unknown is 2. The standard form is ax² + bx + c = 0, with a ≠ 0. The chapter teaches you four things: how to recognise a quadratic, how to find its roots, how to judge the nature of the roots without solving, and how to use the sum and product of roots.

The questions are short and formula-driven. Most can be done in under a minute if you know the right shortcut. You will be asked to find roots, to form an equation from given roots, to find an unknown constant so that roots are equal or real, or to find values like α² + β² from the coefficients.

This chapter connects to the rest of Paper 3 in a few ways. It builds on the algebra you use in linear equations and indices. The same ideas appear again in word problems on cost, revenue and profit, where a quadratic often models the situation. Clean algebra here also helps in calculus topics such as finding maxima and minima, and in the speed you show across the paper.

Quadratic equations are a high-scoring chapter because the questions are predictable and the methods are fixed. With no negative marking, you can attempt every question, and here a few formulas give you quick, reliable marks. A student who learns the discriminant rules and the sum and product relations can answer many questions by direct substitution, saving time for harder chapters in the one-hour paper.

Quadratic Equations: topics in the order to study them

  1. 1Introduction to Quadratic EquationsYou need the standard form, the meaning of a root and the condition a ≠ 0 before you can solve anything.
  2. 2Solving Quadratic EquationsFactorisation and the quadratic formula are the core skills, and the next two topics build on them.
  3. 3Nature of Roots and DiscriminantThe discriminant comes from the quadratic formula, so it is easy to understand once you can solve equations.
  4. 4Sum and Product of RootsThis is the shortcut topic. It uses the coefficients directly and often combines with the discriminant in one question.

How to prepare Quadratic Equations

Aim for speed with accuracy. Each question should take about a minute, so practise methods, not long working.

  1. Write the standard form ax² + bx + c = 0 and learn to rearrange any equation into it, moving all terms to one side.
  2. Practise factorisation with simple numbers until you can split the middle term in a few seconds. Use the formula when factors are not obvious.
  3. Learn the discriminant D = b² − 4ac and its three cases: D > 0 gives two distinct real roots, D = 0 gives equal real roots, D < 0 gives no real roots.
  4. Memorise sum of roots = −b/a and product of roots = c/a. Practise forming an equation: x² − (sum)x + (product) = 0.
  5. Learn to rewrite expressions using sum and product, for example α² + β² = (α + β)² − 2αβ.
  6. Solve MCQs under time. When stuck, substitute the four options into the equation. It is often the fastest check.
  7. Keep an error list of sign mistakes and revise it before the exam.

Common mistakes in Quadratic Equations

  • Applying formulas without first writing the equation in standard form.

    Fix: Always move everything to one side and arrange as ax² + bx + c = 0 before reading a, b and c.

  • Making sign errors in −b/a and in the formula.

    Fix: Write the values of a, b and c with their signs first, then substitute with brackets.

  • Writing the equation from roots as x² + (sum)x + (product) = 0.

    Fix: Remember the form x² − (sum)x + (product) = 0. Check by finding the sum from your answer.

  • Calculating b² − 4ac wrongly when b or c is negative.

    Fix: Compute b² and 4ac separately, keeping signs, then subtract. Remember b² is never negative.

  • Forgetting to check both roots or dropping the ± in the formula.

    Fix: Find both values and check which options match. Verify by substituting a root back.

  • Solving fully when only sum, product or nature of roots is asked.

    Fix: Read the question first. If it asks about nature, use D. If it asks about sum or product, use the coefficients.

Last-day revision: Quadratic Equations

  • Standard form: ax² + bx + c = 0, with a ≠ 0.
  • Roots are the values of x that satisfy the equation.
  • Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a.
  • Discriminant D = b² − 4ac.
  • D > 0: two distinct real roots. D = 0: equal real roots. D < 0: no real roots.
  • For rational coefficients, if D is a perfect square, the roots are rational.
  • Sum of roots = −b/a.
  • Product of roots = c/a.
  • Equation from roots: x² − (sum)x + (product) = 0.
  • α² + β² = (α + β)² − 2αβ.
  • Roots are reciprocals of each other when c = a.
  • Roots are equal in magnitude and opposite in sign when b = 0.

Quadratic Equations practice questions

Quadratic Equations in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Quadratic Equations: frequently asked questions

What is the quickest way to solve a quadratic equation in the exam?

Try factorisation first if the numbers are small. If it does not factorise quickly, use the quadratic formula. You can also test the four options by substitution, which often takes a few seconds.

What does the discriminant tell me?

The discriminant D = b² − 4ac tells you the nature of the roots without solving. If D is positive, there are two distinct real roots. If D is zero, the roots are real and equal. If D is negative, there are no real roots.

Do I need to memorise the sum and product formulas?

Yes. The sum of roots is −b/a and the product is c/a. They give direct answers to many MCQs and help you form equations from given roots quickly.

How much time should I give this chapter?

The concepts are few, so a focused few days of practice is usually enough for most students. Spend more time on MCQ practice than on reading, because speed and sign accuracy decide your marks.