Question

Find the roots of the following equation, if they exist, by applying the quadratic formula:

x² – 2ax + (a² – b²) = 0 

Answer 1

Given: 

x² – 2ax + (a² – b²) = 0 

On comparing it with Ax² + Bx + C = 0 we get 

A = 1, B = –2a and C = (a² – b²)  

Discriminant D is given by: 

D = B² – 4AC 

= (–2a)² – 4 × 1 × (a² – b²)  

= 4a² – 4a² + 4b² 

= 4b² > 0 

Hence, the roots of the equation are real. 

Roots α and β are given by:  

`α = (-b + sqrt(D))/(2a)`

= `(-(-2a) + sqrt(4)b^2)/(2 xx 1)`

= `(2a + 2b)/2`

= `(2(a + b))/2`

= (a + b)

`β = (-b - sqrt(D))/(2a)`

= `(-(-2a) - sqrt(4)b^2)/(2 xx 1)`

= `(2a - 2b)/2`

= `(2(a - b))/2`

= (a – b) 

Hence, the roots of the equation are (a + b) and (a – b).