Question

Find m if (m – 12) x² + 2(m – 12) x + 2 = 0 has real and equal roots.

Answer 1

(m − 12)x² + 2(m − 12)x + 2 = 0

Comparing the above equation with ax² + bx + c = 0, we get

a = m − 12, b = 2(m − 12), c = 2

∆ = b² − 4ac

b² − 4ac = [2(m − 12)]² − 4 × (m − 12) × 2

= 4(m – 12)² – 8m + 96 

= 4(m² − 24m + 144) − 8m + 96

= 4m² - 96m + 576 − 8m + 96

= 4m² − 104m + 672

The roots of the given quaaratic equation are real and equal.        ... (Given)

∴ b² − 4ac = 0

∴ 4m² - 104m + 672 = 0

∴ m² − 26m + 168 = 0       ...(Dividing by 4)

∴ m − 12m - 14m + 168 = 0

∴ m(m − 12) − 14(m - 12) = 0

∴ (m −12)(m - 14) = 0

∴ m − 12 = 0 or m − 14 = 0

∴ m = 12 or m = 14

If m = 12, m − 12 = 0 and (m − 12)x² = 0

The equation will not be a quadratic one.

∴ m = 12 is unacceptable.

∴ m = 14

∴ The value of m is 14.