Question

If the quadratic equation (a2 – b2)x2 + (b2 – c2)x + (c2 – a2) = 0 has equal roots, then which of the following is true? (A) b2 + c2 = a2 (B) b2 + c2 = 2a2 (C) b2 – c2 = 2a2 (D) a2 = b2 + 2c2

Answer 1

IF THE QUADRATIC EQUATION

Answer 2

Given:

The roots of the equation (a2 – b2)x2 + (b2 – c2)x + (c2 – a2) = 0 are equal

To find:

If the roots of the eqn (a2 – b2)x2 + (b2 – c2)x + (c2 – a2) = 0 are equal, then

Solution:  

From given, we have.

The roots of the eqn (a2 – b2)x2 + (b2 – c2)x + (c2 – a2) = 0 are equal

The condition for the roots of a quadratic equation to be equal is,

b² - 4ac = 0

$\left(\left(b\cdot \:2-c\cdot \:2\right)\right)^2-4\left(a\cdot \:2-b\cdot \:2\right)\left(c\cdot \:2-a\cdot \:2\right)=0\\\\4b^2-16ab+8bc+16a^2+4c^2-16ac=0\\\\16a^2-\left(16b+16c\right)a+4b^2+8bc+4c^2=0$

upon simplifying, we get,

$a_{1,\:2}=\dfrac{-\left(-16b-16c\right)\pm \sqrt{0}}{2\cdot \:16}\\\\a=\frac{b+c}{2}\\\\2a=b+c$

Therefore, 2a = b + c is correct answer.