How to solve quadratic equations in Class 10

First write the equation in the form ax² + bx + c = 0, with a ≠ 0. Read the sign of each coefficient as part of the number. Then check whether the expression factorises easily. When the factors are hard to spot, use the quadratic formula.

Example 1. Factorise x² − 9x + 20 = 0

Find two numbers that multiply to 20 and add to −9. The numbers are −4 and −5, so the equation becomes (x − 4)(x − 5) = 0. At least one bracket must equal zero, giving x = 4 or x = 5. Check the first root by substituting it back: 16 − 36 + 20 = 0.

Example 2. Use the formula for 2x² − 3x − 1 = 0

Here a = 2, b = −3 and c = −1. The discriminant b² − 4ac equals 9 − 4(2)(−1), which is 17. A positive discriminant gives two distinct real roots.

Substitute into x = (−b ± √(b² − 4ac)) / (2a). The solutions are x = (3 + √17) / 4 and x = (3 − √17) / 4.

Catch the sign mistake before it costs a mark

In the formula, −b is the opposite of b. Since b = −3, the numerator starts with +3. The entire numerator is divided by 2a. If you’re writing by hand, draw the fraction bar across both the number and the square-root term.

Try three questions

1. Solve x² − 10x + 21 = 0 by factorisation.
2. Use the discriminant to decide how many distinct real roots 3x² + 2x + 1 = 0 has.
3. A rectangular noticeboard has an area of 70 cm². Its length is 3 cm more than its width. Find both dimensions.

Check the answers

1. (x − 3)(x − 7) = 0, so x = 3 or 7.
2. The discriminant is 4 − 12 = −8, so there are no real roots.
3. Let the width be w. Then w(w + 3) = 70, so (w − 7)(w + 10) = 0. The width is 7 cm and the length is 10 cm. A physical width must be positive.

For six more problems, print the Class 10 quadratic equations worksheet. Use the Class 10 worksheet library to switch to a different revision topic.