Write the condition to be satisfied for which equations ax^2 + 2bx + c = 0 and bx^2 - 2 √acx + b = 0 have equal roots.

Question

Write the condition to be satisfied for which equations ax² + 2bx + c = 0 and \(b\text{x}^2-2\sqrt{ac}\text{x}+b=0\) have equal roots.

Answer 1

Given the roots of both the equations are real 

For first equation ax² + 2bx + c = 0 

Its discriminant; d ≥ 0

D = b² – 4ac

D = (2b)² – 4 × a × c ≥ 0 

4b² ≥ b² ≥ ac …1 

For second equation

bx² - 2 \(\sqrt{ac}\) x + b = 0

d = b² – 4ac ≥ 0

\(=(2\sqrt{ac})^2-4\) b xb ≥ 0

= 4ac – 4 b² ≥ \(0\)

From 1 and 2 we get only one case where b² = ac