Solve the following quadratic equation by factorization:
$x^2+(a+\frac{1}{a})x+1=0$
Given:
Given quadratic equation is $x^2+(a+\frac{1}{a})x+1=0$.
To do:
We have to solve the given quadratic equation.
Solution:
$x^2+(a+\frac{1}{a})x+1=0$
To factorise $x^2+(a+\frac{1}{a})x+1=0$, we have to find two numbers $m$ and $n$ such that $m+n=a+\frac{1}{a}$ and $mn=1(1)=1$.
If $m=a$ and $n=\frac{1}{a}$, $m+n=a+\frac{1}{a}$ and $mn=a\times\frac{1}{a}=1$.
$x^2+(a+\frac{1}{a})x+1=0$
$x^2+ax+\frac{1}{a}x+1=0$
$x(x+a)+\frac{1}{a}(x+a)=0$
$(x+a)(x+\frac{1}{a})=0$
$x+a=0$ or $x+\frac{1}{a}=0$
$x=-a$ or $x=-\frac{1}{a}$
The values of $x$ are $-a$ and $-\frac{1}{a}$.
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