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Quantitative Aptitude · Equations

Quadratic Equations and Their Roots for CA Foundation

Updated 1 October 2026 · Fact-checked

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Solve it by factorising or by the formula x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant D = b² − 4ac tells you if the roots are real and distinct, equal, or not real.

Understand Quadratic Equations and Their Roots

A quadratic equation is an equation where the highest power of the unknown is 2. Its standard form is ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. If a = 0, the x² term disappears and it becomes a linear equation.

A root is a value of x that makes the equation true. A quadratic has two roots, which may be different, equal, or not real. Each root, when put back in the equation, gives 0 on the left side.

There are two main ways to find the roots. In factorisation, you split the middle term so the expression becomes a product of two brackets, then set each bracket to zero. This works because if a product is zero, at least one factor is zero. The quadratic formula works for every quadratic, even when factorising is hard.

The discriminant is D = b² − 4ac. It is the part under the square root in the formula, so it decides what kind of roots you get. If D > 0, there are two distinct real roots. If D = 0, there are two equal real roots. If D < 0, the roots are not real. For a, b, c rational, if D is also a perfect square, the roots are rational.

Key formulas to remember

Standard form
ax² + bx + c = 0, a ≠ 0
Move every term to one side and arrange in powers of x before you read a, b, c.
Quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
Works for every quadratic. The ± gives the two roots.
Discriminant
D = b² − 4ac
Only the sign of D and whether it is zero or a perfect square matter for nature of roots.
Nature of roots
D > 0: real and distinct; D = 0: real and equal; D < 0: not real
For rational a, b, c: D a positive perfect square means rational roots; D positive but not a perfect square means irrational roots.
Sum and product of roots
α + β = −b ÷ a; αβ = c ÷ a
Useful for checking your answers quickly.
Equation from roots
x² − (α + β)x + αβ = 0
Use when the roots are given and you need the equation.

How to solve Quadratic Equations and Their Roots questions

Use this method for any question on solving quadratics or finding the nature of roots.

  1. 1Bring all terms to one side so the equation reads ax² + bx + c = 0. Clear fractions or brackets first.
  2. 2Write down a, b and c with their signs. A missing term means its coefficient is 0.
  3. 3If the question asks only about the nature of roots, compute D = b² − 4ac and stop there.
  4. 4Compare D with 0, and check if it is a perfect square if rationality matters. State the nature of the roots.
  5. 5If the question asks for roots, try factorisation: find two numbers whose product is a × c and whose sum is b. Split the middle term and group.
  6. 6If no such pair is quick to find, use the formula x = (−b ± √D) ÷ 2a.
  7. 7Put one root back in the equation, or check sum = −b ÷ a and product = c ÷ a, to confirm.

Quickest way: Option testing with sum and product

When to use it: Use when the MCQ gives numerical roots as options and the equation has integer coefficients.

  1. Find −b ÷ a (the sum) and c ÷ a (the product) from the equation.
  2. Scan the options for the pair that adds to the sum. Reject the rest at once.
  3. Confirm the product of that pair. Only one option should fit.
  4. For nature of roots, compute only D. Do not solve the equation.
  5. If D is large or messy, skip and return later. Wrong answers cost 0.25 marks.

Common mistakes in Quadratic Equations and Their Roots

  • Reading a, b, c before moving all terms to one side.

    Students take numbers straight from an equation like x² = 5x − 6.

    Fix: Rewrite as x² − 5x + 6 = 0 first. Then a = 1, b = −5, c = 6.

  • Dropping the sign of b in −b or in b².

    Rushing with negative coefficients.

    Fix: Write b in brackets. For b = −5, −b = 5 and b² = 25.

  • Getting the sign of 4ac wrong when a or c is negative.

    Two minus signs are confused.

    Fix: Substitute with brackets. If ac is negative, then −4ac is positive, e.g. a = 1, c = −6 gives D = b² + 24.

  • Dividing both sides by x and losing a root.

    Students cancel x in an equation like x² = 3x.

    Fix: Move everything to one side and factorise: x(x − 3) = 0, so x = 0 or x = 3.

  • Forgetting that a perfect-square D gives rational roots only when a, b, c are rational.

    The rule is remembered without its condition.

    Fix: Check that coefficients are rational first, then test whether D is a perfect square.

  • Setting each factor equal to the constant, e.g. in (x − 2)(x − 3) = 6, wrongly setting x − 2 = 6 and x − 3 = 6 to get x = 8 and x = 9.

    The zero product rule is applied when the right side is not zero.

    Fix: Neither 8 nor 9 satisfies (x − 2)(x − 3) = 6. Expand to x² − 5x + 6 = 6, move 6 across to get x² − 5x = 0, and factorise: x(x − 5) = 0, so x = 0 or 5.

Worked examples

Example 1

The roots of the equation x² − 7x + 12 = 0 are: (a) 3 and 4 (b) −3 and −4 (c) 2 and 6 (d) 1 and 12

Show the solution
  1. Here a = 1, b = −7, c = 12.
  2. Find two numbers with product 12 and sum −7. These are −3 and −4.
  3. Split the middle term: x² − 3x − 4x + 12 = 0.
  4. Group: x(x − 3) − 4(x − 3) = 0, so (x − 3)(x − 4) = 0.
  5. Hence x = 3 or x = 4.
  6. Check: sum = 7 = −b ÷ a and product = 12 = c ÷ a.

Answer: (a) 3 and 4

Example 2

The nature of the roots of 2x² − 4x + 5 = 0 is: (a) real and distinct (b) real and equal (c) not real (d) rational and distinct

Show the solution
  1. Here a = 2, b = −4, c = 5.
  2. D = b² − 4ac = (−4)² − 4(2)(5) = 16 − 40 = −24.
  3. D < 0, so the roots are not real.
  4. Options (a), (b) and (d) all need real roots, so they are out.

Answer: (c) not real

Example 3

If the roots of x² − 6x + k = 0 are real and equal, then k is: (a) 3 (b) 6 (c) 9 (d) 36

Show the solution
  1. Here a = 1, b = −6, c = k.
  2. Equal roots need D = 0.
  3. D = (−6)² − 4(1)(k) = 36 − 4k.
  4. Set 36 − 4k = 0, so k = 9.
  5. Check: x² − 6x + 9 = (x − 3)², with equal roots x = 3.

Answer: (c) 9

Exam tips

  • For nature-of-roots questions, compute D only. It saves time and avoids square-root errors.
  • When a question has a parameter like k, set up D = 0 for equal roots, D > 0 for distinct real roots and D < 0 for non-real roots.
  • Test the options by sum and product before you start factorising.
  • Always move terms to one side first. Many MCQs give equations in disguised form.
  • If an equation takes more than about a minute, mark it and move on. Wrong answers lose 0.25 marks.

Practice questions from Equations

Quadratic Equations and Their Roots: frequently asked questions

What is the discriminant of a quadratic equation?

The discriminant is D = b² − 4ac for the equation ax² + bx + c = 0. Its sign shows the nature of the roots. It is the expression under the square root in the quadratic formula.

When should I use the quadratic formula instead of factorisation?

Use factorisation when you can quickly find two numbers with product ac and sum b. If that takes too long, or D is not a perfect square, use the formula. The formula works for every quadratic.

What if the discriminant is negative?

The equation has no real roots. In CA Foundation MCQs, you simply state that the roots are not real. You do not need to find them.

Can a quadratic equation have only one root?

When D = 0, the two roots are equal, so there is one distinct value that is counted twice. For example, x² − 6x + 9 = 0 gives x = 3 twice.