Solve $4x^{2}$ - 16x + 15 = 0.

Analyze the question: We need to find the roots of the quadratic equation 4x² − 16x + 15 = 0.

Concept: Use factorisation by splitting the middle term. Find two numbers whose product is a × c = 4 × 15 = 60 and whose sum is −16.

Solution:

The numbers are −6 and −10, since (−6)(−10) = 60 and −6 + (−10) = −16.

4x² − 6x − 10x + 15 = 0

Group the terms: 2x(2x − 3) − 5(2x − 3) = 0

(2x − 3)(2x − 5) = 0

So:

• 2x − 3 = 0 gives x = 3/2 = 1½
• 2x − 5 = 0 gives x = 5/2 = 2½

Check: Let x = 1½. Then 4(2.25) − 16(1.5) + 15 = 9 − 24 + 15 = 0 ✓. Let x = 2½. Then 4(6.25) − 40 + 15 = 25 − 40 + 15 = 0 ✓.

Answer: x = 1½ or x = 2½, which is option B.

Tip: For a quadratic with positive a, negative b and positive c, both roots are positive. This means you can immediately eliminate options A, C and D, which have negative roots. You can also check with the sum of roots: −b/a = 16/4 = 4, and 1½ + 2½ = 4 ✓.