Class 10 · Mathematics · FBISE Islamabad

Chapter 1: Quadratic Equations

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Method

1 4 mkMethod

Solve the quadratic equation x² − 5x + 6 = 0 by factorisation.

STEP 1 Find two numbers whose product is 6 and whose sum is −5: they are −2 and −3.

STEP 2 x² − 2x − 3x + 6 = 0

STEP 3 x(x − 2) − 3(x − 2) = 0

STEP 4 (x − 2)(x − 3) = 0

STEP 5 x = 2 or x = 3

Answer x = 2, 3

Common mistake

Write the middle-term split as a separate line. Jumping straight to the brackets loses a working mark.

2 5 mkMethod

Solve 2x² + 3x − 5 = 0 using the quadratic formula.

a = 2b = 3c = −5Find x

STEP 1 x = [−b ± √(b² − 4ac)] ÷ 2a

STEP 2 b² − 4ac = 9 − 4(2)(−5) = 9 + 40 = 49

STEP 3 x = (−3 ± 7) ÷ 4

STEP 4 x = 4/4 = 1 or x = −10/4 = −2.5

Answer x = 1 or x = −2.5

Common mistake

Watch the sign of c. With c negative, −4ac becomes positive and is added.

Concept

3 4 mkConcept

What is the discriminant? What does its value tell you about the roots of ax² + bx + c = 0?

DEFINE The discriminant is D = b² − 4ac.

CASE 1 If D > 0, the roots are real, distinct and unequal.

CASE 2 If D = 0, the roots are real and equal.

CASE 3 If D < 0, the roots are imaginary (complex conjugates).

Answer D = b² − 4ac; its sign decides whether the roots are unequal, equal or imaginary.

Common mistake

All three cases must be listed. Two out of three gets partial credit only.

Numerical

4 5 mkNumerical

The sum of a number and its reciprocal is 5/2. Find the number.

STEP 1 Let the number be x, so x + 1/x = 5/2

STEP 2 Multiply through by 2x: 2x² + 2 = 5x

STEP 3 2x² − 5x + 2 = 0

STEP 4 (2x − 1)(x − 2) = 0

STEP 5 x = 1/2 or x = 2

Answer The number is 2 or 1/2

Common mistake

Both values are valid answers — they are reciprocals of each other. Do not discard one.

This sheet was checked against the FBISE Islamabad 2024 marking scheme.