Question

Find the value of α for which the equation (α – 12)x² + 2(α – 12)x + 2 = 0 has equal roots.

Answer 1

Given: 

(α – 12)x² + 2(α – 12)x + 2 = 0

Here, 

a = (α – 12), b = 2(α – 12) and c = 2 

It is given that the roots of the equation are equal; therefore, we have 

D = 0 

⇒ (b² – 4ac) = 0 

⇒ {2(α – 12)}² – 4 × (α – 12) × 2 = 0 

⇒ 4(α² – 24α + 144) – 8α + 96 = 0 

⇒ 4α² – 96α + 576 – 8α + 96 = 0 

⇒ 4α² – 104α + 672 = 0 

⇒ α² – 26α + 168 = 0 

⇒ α² – 14α – 12α + 168 = 0 

⇒ α(α – 14) – 12(α – 14) = 0 

⇒ (α – 14)(α – 12) = 0 

∴ α = 14 or α = 12 

If the value of α is 12, the given equation becomes non-quadratic.

Therefore, the value of α will be 14 for the equation to have equal roots.