Solve the following quadratic equation by factorization method: 1/(a+b+x) = 1/a + 1/b +1/x , a and b is not equal to zero

factorization method: 1/(a+b+x) = 1/a + 1/b +1/x

Q) Solve the following quadratic equations by factorization method:

$\frac{1}{a+b+x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{x},a+b\ne0$

Ans:

$\frac{1}{a+b+x}-\frac{1}{x}=\frac{1}{a}+\frac{1}{b}$

$\frac{x-(a+b+x)}{x(a+b+x)}=\frac{a+b}{ab}$

$\frac{-(a+b)}{x(a+b+x)}=\frac{a+b}{ab}$

$-ab(a+b)=(a+b)x(a+b+x)$

$-ab\cancel{(a+b)}=\cancel{(a+b)}x(a+b+x)$

$-ab=x(a+b+x)$

$x(a+b+x)+ab=0$

$ax+bx+x^2+ab=0$

$x^2+ax+bx+ab=0$

$x(x+a)+b(x+a)=0$

$(x+a)(x+b)=0$

$x+a=0\implies x=-a, ~or,$

$x+b=0\implies x=-b$

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