factorisation of a²x - b²x
a) (ax+bx)(ax-bx)
b) (ax-bx)²
c) (ax+bx) (a²-b²)
d) (a*x-b*x)(a²+b²)
Answers
SOLUTION :
Given : (x - a)/(x - b) + ( x - b)/( x - a) = a/b + b/a
[(x - a)² + (x -b)² ] / [(x - a)(x - b)] = [(a² + b²)/ab]
[ By taking LCM]
(x² - 2a + a² + x² - 2b + b²) /(x² - ax - bx + ab) = [(a² + b²)/ab]
ab(x² - 2a + a² + x² - 2b + b²) = (a² + b²)(x² - ax - bx + ab)
ab(x² - 2a + a² + x² - 2b + b²) = (a² + b²)(x² - ax - bx + ab)
ab(x² + x² - 2a - 2b + a² + b²) = (a² + b²)(x² - ax - bx + ab)
ab[ 2x² -2(a + b)x +(a² + b²)] = (a² +b²)x² -(a² +b²)(a +b)x +(a² + b²)ab
[ 2abx² -2abx(a + b) + ab(a² + b²)] = (a² +b²)x² -(a² +b²)(a +b)x +(a² + b²)ab
[2abx² - 2abx(a + b) + ab(a² + b²)] = (a² +b²)x² - (a² +b²)(a +b)x + (a² + b²)ab
[2abx² - (a² +b²)x² - 2abx(a + b) + (a² +b²)(a +b)x] = (a² + b²)ab - (a² + b²)ab
[(2ab - a² - b²)x² - (a +b)(2ab - a² - b²)x = 0
(a² + b² -2ab)x² - (a + b)(a² + b² - 2ab)x = 0
(a - b)²x² - (a + b)(a - b)² x =0
[We know that , (a - b)² = a² + b² - 2ab]
(a - b)² [x² - (a + b) x ] = 0
x² - (a + b)x = 0
x[ x - (a + b)] =0
x = 0 or x - (a + b) = 0
x = 0 or x = a + b
Hence, the roots of the quadratic equation (x - a)/(x - b) + ( x - b)/( x - a) = a/b + b/a are 0 & (a + b) .
HOPE THIS ANSWER WILL HELP YOU….