🏹 Class 10 · Algebra

Quadratic Equations for Class 10: notes, worked examples and practice

Q: What is the quadratic formula?

x = (−b ± √(b² − 4ac)) ÷ 2a for the equation ax² + bx + c = 0.

Q: What does the discriminant tell us?

If b² − 4ac is positive there are two real roots, if it is zero there is one repeated root, and if it is negative there are no real roots.

Q: When should I factorise instead of using the formula?

When the equation factorises easily. Otherwise the formula always works.

Q: How can I check my roots?

Put them back into the equation, or check that they add to −b/a and multiply to c/a.

Q: Which board is this quadratic equations page for?

It covers the ideas taught in CBSE, ICSE and state-board maths. Chapter names and order differ by board and textbook edition, so follow your own textbook for the exact order.

A quadratic equation has an x² term. It can have two solutions, one or none, and the discriminant tells you which before you solve.

Updated 21 October 2026. Every worked example and answer on this page is checked by automated tests.

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Notes

Quadratic Equations: what you need to know

Chapter names and order differ by board and textbook edition, so follow your own textbook for the order.

Standard form

A quadratic equation is written ax² + bx + c = 0 with a ≠ 0. Always rearrange it into this form first.

Solving by factorisation

Factorise the left side and set each factor to zero. x² − 7x + 12 = (x − 3)(x − 4) = 0, so x = 3 or x = 4.

The quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a. It works for every quadratic equation.

The discriminant

D = b² − 4ac. If D > 0 there are two distinct real roots. If D = 0 there are two equal real roots. If D < 0 there are no real roots.

Sum and product of roots

The roots add up to −b/a and multiply to c/a. Use this to check your answers.

Formulas and rules to remember

ax² + bx + c = 0, a ≠ 0
x = (−b ± √(b² − 4ac)) ÷ 2a
D = b² − 4ac
Sum of roots = −b/a, product = c/a

Solved examples

Example 1

Solve x² − 7x + 12 = 0.

Find two numbers with product 12 and sum −7: −3 and −4.
(x − 3)(x − 4) = 0.
x = 3 or x = 4.

Answer: x = 3 or x = 4

Example 2

Solve 2x² + x − 6 = 0 using the formula.

a = 2, b = 1, c = −6.
D = 1² − 4 × 2 × (−6) = 1 + 48 = 49.
x = (−1 ± 7) ÷ 4, so x = 3/2 or x = −2.

Answer: x = 3/2 or x = −2

Example 3

Find the nature of the roots of x² + 4x + 5 = 0.

a = 1, b = 4, c = 5.
D = 16 − 20 = −4.
D is negative, so there are no real roots.

Answer: −4, so there are no real roots

Example 4

The product of two consecutive positive integers is 132. Find them.

Let them be n and n + 1: n(n + 1) = 132.
n² + n − 132 = 0, which factorises as (n − 11)(n + 12) = 0.
n = 11, so the numbers are 11 and 12.

Answer: 11 and 12

Avoid these

Common mistakes

Forgetting the ± in the formula

Both the plus and minus give a root. Drop one and you lose a solution.

Sign slip with −b

If b = −7 then −b = +7, so the formula starts with 7 ± √D, not −7 ± √D.

Dividing by 2 instead of 2a

The whole numerator is divided by 2a.

Not rearranging to standard form

Move everything to one side so that the other side is 0 before you factorise.

10 practice questions with answers

Try each question on paper first, then tap it to check your answer. The questions are generated by code and each answer is checked.

Find the discriminant of 3x² − 5x + 2 = 0.
Answer: 1
Solve x² − 4x + 3 = 0. The roots are
Answer: 1 and 3
Solve x² + 2x + 1 = 0. The roots are
Answer: −1 (equal roots)
Find the discriminant of x² + 5x − 1 = 0.
Answer: 29
Find the discriminant of 2x² + 1 = 0.
Answer: -8
Solve x² + 2x − 35 = 0. The roots are
Answer: −7 and 5
Find the discriminant of 2x² − 6x − 4 = 0.
Answer: 68
Solve x² − 5x − 14 = 0. The roots are
Answer: −2 and 7
Solve x² − 7x + 12 = 0. The roots are
Answer: 3 and 4
Solve x² − 4x + 4 = 0. The roots are
Answer: 2 (equal roots)

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FAQ

Quadratic Equations Class 10: questions students ask

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) ÷ 2a for the equation ax² + bx + c = 0.

What does the discriminant tell us?

If b² − 4ac is positive there are two real roots, if it is zero there is one repeated root, and if it is negative there are no real roots.

When should I factorise instead of using the formula?

When the equation factorises easily. Otherwise the formula always works.

How can I check my roots?

Put them back into the equation, or check that they add to −b/a and multiply to c/a.

Which board is this quadratic equations page for?

It covers the ideas taught in CBSE, ICSE and state-board maths. Chapter names and order differ by board and textbook edition, so follow your own textbook for the exact order.

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