Question
Solve the following quadratic equations by factorization:
$\frac{\text{a}}{\text{x}-\text{b}}+\frac{\text{b}}{\text{x}-\text{a}}=2$
$\frac{\text{a}}{\text{x}-\text{b}}+\frac{\text{b}}{\text{x}-\text{a}}=2$
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Answer
$\frac{\text{a}}{\text{x}-\text{b}}+\frac{\text{b}}{\text{x}-\text{a}}=2$
$\Rightarrow\frac{\text{a}(\text{x}-\text{a})+\text{b}(\text{x}-\text{b})}{(\text{x}-\text{a})(\text{x}-\text{b})}=2$
$\Rightarrow a x-a^2+b x-b^2=2 x^2-2 a x-2 b x+2 a b$
$\Rightarrow 2 x^2-2 a x-a x-2 b x-b x+a^2+b^2+2 a b=0$
$\Rightarrow 2 x^2-3 x(a+b)+(a+b)^2=0$
$\Rightarrow 2 x^2-2 x(a+b)-x(a+b)+(a+b)^2=0$
$\Rightarrow 2 x[x-(a+b)]-(a+b)[x-(a+b)]=0$
$\Rightarrow[2 x-(a+b)][(x-a+b)]=0$
$\Rightarrow x=\frac{a+b}{2} \text { or } x=a+b$
$\Rightarrow\text{x}=\frac{\text{a}+\text{b}}{2}$ or $x = a + b$
$\Rightarrow\frac{\text{a}(\text{x}-\text{a})+\text{b}(\text{x}-\text{b})}{(\text{x}-\text{a})(\text{x}-\text{b})}=2$
$\Rightarrow a x-a^2+b x-b^2=2 x^2-2 a x-2 b x+2 a b$
$\Rightarrow 2 x^2-2 a x-a x-2 b x-b x+a^2+b^2+2 a b=0$
$\Rightarrow 2 x^2-3 x(a+b)+(a+b)^2=0$
$\Rightarrow 2 x^2-2 x(a+b)-x(a+b)+(a+b)^2=0$
$\Rightarrow 2 x[x-(a+b)]-(a+b)[x-(a+b)]=0$
$\Rightarrow[2 x-(a+b)][(x-a+b)]=0$
$\Rightarrow x=\frac{a+b}{2} \text { or } x=a+b$
$\Rightarrow\text{x}=\frac{\text{a}+\text{b}}{2}$ or $x = a + b$
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