Question

Solve by factorization method: - \[x+\dfrac{1}{x}=11\dfrac{1}{11}\].

Answer 1

Hint: First convert the given mixed fraction into improper fraction by using the conversion: - \[a\dfrac{b}{c}=\dfrac{\left( a\times c \right)+b}{c}\]. Now, take L.C.M in the L.H.S and cross – multiply the terms to form a quadratic equation in x. Use the middle term split method to write the obtained quadratic equation in the form: - \[\left( x-a \right)\left( x-b \right)=0\], where ‘a’ and ‘b’ are the factors of the equation. Finally, substitute each term equal to zero and obtain the required value of x.

Complete step-by-step answer:
Here, we have been provided with the equation: - \[x+\dfrac{1}{x}=11\dfrac{1}{11}\] and we have to solve it using the factorization method. So, we have to solve it using the middle term split method.
Now, converting the given mixed fraction into improper fraction y using the relation: - \[a\dfrac{b}{c}=\dfrac{\left( a\times c \right)+b}{c}\], we get,
\[\Rightarrow 11\dfrac{1}{11}=\dfrac{\left( 11\times 11 \right)+1}{11}=\dfrac{122}{11}\]
So, the equation becomes,
\[\Rightarrow x+\dfrac{1}{x}=\dfrac{122}{11}\]
Taking L.C.M in the L.H.S, we get,
\[\Rightarrow \dfrac{{{x}^{2}}+1}{x}=\dfrac{122}{11}\]
By cross – multiplication we get,
\[\begin{align}
  & \Rightarrow 11{{x}^{2}}+11=122x \\
 & \Rightarrow 11{{x}^{2}}-122x+11=0 \\
\end{align}\]
Now, splitting the middle term such that its product is equal to \[11\times 11=121\], we get,
\[\begin{align}
  & \Rightarrow 11{{x}^{2}}-121x-x+11=0 \\
 & \Rightarrow 11x\left( x-11 \right)-1\left( x-11 \right)=0 \\
\end{align}\]
Taking common terms together, we get,
\[\Rightarrow \left( x-11 \right)\left( 11x-1 \right)=0\]
Substituting each term equal to 0, we get,
\[\Rightarrow x=11\] or \[x=\dfrac{1}{11}\]
Hence, the two solutions of the given equations are 11 and \[\dfrac{1}{11}\]. These are also called the factors of the given equation.

Note: One may note that we can solve the obtained quadratic equation by the help of the discriminant method given by the formula: - \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]. But as you can see that we have been asked in the question to solve it using factorization method, that means we have to form factors such as \[\left( x-a \right),\left( x-b \right)\] and substitute them equal to zero one – by – one. Here, ‘a’ and ‘b’ are the factors of the quadratic equation. You must remember the process of the middle term split method to solve the question.