Question

Solve the following quaditic equations for x: x^2+(a/a+b + a+b/a) x+1=0

Answer 1

Given equation,

$\sf{x {}^{2} + ( \frac{a}{a + b} + \frac{a + b}{a} )x + 1 = 0 }$

Now,

The constant term can be written as:

$\sf{1 = \frac{a}{a + b}. \frac{a + b}{a} }$

By splitting the middle term,

$\sf{x {}^{2} + \frac{a}{a + b}x + \frac{a + b}{a}x + \frac{a}{a + b}. \frac{a + b}{a} = 0 } \\ \\ \implies \: \sf{x(x + \frac{a}{a + b}) + \frac{a + b}{a}(x + \frac{a}{a + b}) = 0 } \\ \\ \implies \: \sf{(x + \frac{a + b}{a})(x + \frac{a}{a + b}) = 0 } \\ \\ \implies \: \boxed{ \sf{x = \frac{ - a}{a + b} \: \: or \: \: \frac{ - (a + b)}{a} }}$

The roots of the equation are -a/a+b and -(a+b)/a