Question

How do you find the factors of \[{{x}^{2}}+11x=2\]?

Answer 1

Hint: This type of question is based on the concept of factorisation. We can solve this question with the help of factorisation of quadratic equations. This type of equation is normally solved by the formula \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]. Then after obtaining the value of ‘x’, the factors of x will be \[\left( x-\dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a} \right)\] and \[\left( x-\dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a} \right)\]. On further calculations, we get the factors of ‘x’.

Complete step by step answer:
According to the question, we are asked to find the factors of the given equation \[{{x}^{2}}+11x=2\].
We have been given the equation is \[{{x}^{2}}+11x=2\]
Let us subtract 2 from both the sides of the equation.
We get,
\[\Rightarrow {{x}^{2}}+11x-2=2-2\]
\[\therefore {{x}^{2}}+11x-2=0\] ------(1)
 We now have to find \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] --------(2)
Comparing this with equation (1), we get,
a=1, b=11 and c=-2
Now, we have to substitute these values in equation (2).
\[x=\dfrac{-11\pm \sqrt{{{\left( 11 \right)}^{2}}-4\left( 1 \right)\left( -2 \right)}}{2\left( 1 \right)}\]
On further simplification, we get,
\[x=\dfrac{-11\pm \sqrt{121-4\left( -2 \right)}}{2}\]
\[\Rightarrow x=\dfrac{-11\pm \sqrt{121+8}}{2}\]
\[\therefore x=\dfrac{-11\pm \sqrt{129}}{2}\]
Now consider the values of ‘x’ separately.
The values of x are \[x=\dfrac{-11+\sqrt{129}}{2}\] and \[x=\dfrac{-11-\sqrt{129}}{2}\]
Therefore, the factors are \[\left( x-\dfrac{-11-\sqrt{129}}{2} \right)\] and \[\left( x-\dfrac{-11+\sqrt{129}}{2} \right)\].
On further simplification, we get,
\[\left( x+\dfrac{11+\sqrt{129}}{2} \right)\] and \[\left( x-\dfrac{-11+\sqrt{129}}{2} \right)\].
Therefore, the factors are \[\left( x+\dfrac{11+\sqrt{129}}{2} \right)\] and \[\left( x-\dfrac{-11+\sqrt{129}}{2} \right)\].
Hence, the factors of \[{{x}^{2}}+11x=2\] is \[\left( x+\dfrac{11+\sqrt{129}}{2} \right)\] and \[\left( x-\dfrac{-11+\sqrt{129}}{2} \right)\].

Note:
Whenever you get this type of problem, we should always try to make the necessary calculations in the given equation to get the final of x. We should then find the factors as required. We should also avoid calculation mistakes based on sign conventions. Using quadratic formula is the only way to find the factors in this question. Calculation mistakes should be avoided to get the accurate answer.