Question

Solve:
\[{{x}^{2}}+x-132\]

Answer 1

Hint: Use the middle term split method to factorize the given quadratic polynomial \[{{x}^{2}}+x-132\] . Split \[+x\] into two terms in such a way that their sum equals \[+x\] and product equals \[-132{{x}^{2}}\]. For this process, find the prime factors of \[-132\times 1\] and combine them in such a manner so that our condition is satisfied. Finally, take the common terms together and write \[{{x}^{2}}+x-132\] as a product of two terms given as \[(x-m)(x-n)\]. Here, ‘\[m\]’ and ‘\[n\]’ are called zeroes of the polynomial.

Complete step by step answer:
 We have been asked to factorize the quadratic polynomial by splitting the middle term: \[{{x}^{2}}+x-132\]. It says that we have to split the middle term which is \[+x\] into two terms such that their sum is \[+x\] and product is \[-132{{x}^{2}}\].
Prime factors of 132 is \[2\times 2\times 3\times 11\].
Now, we have to group these factors so that the conditions of the middle term split method are satisfied as discussed above. So, we have,
a) \[12x-11x=x\]
b) \[(12x)(-11x)=-132{{x}^{2}}\]
Hence, both the conditions of the middle term split method are matched. So, the given quadratic polynomial can be written as: -
\[\begin{align}
  & \Rightarrow {{x}^{2}}+x-132 \\
 & \Rightarrow {{x}^{2}}+12x-11x-132 \\
 & \Rightarrow x(x+12)-11(x+12) \\
\end{align}\]
Taking common from both terms;
\[\begin{align}
  & \Rightarrow x(x+12)-11(x+12) \\
 & \therefore (x+12)(x-11) \\
\end{align}\]
Hence, \[(x+12)(x-11)\] is the factored form of the given quadratic polynomial.

Note:
Here, we can also use the discriminant method to get the factored form of the given quadratic expression. What we can do is, first we will substitute the given expression equal to \[0\] and then we will solve the equation by using the quadratic formula: \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]to get the two values of \[x\]. Now, we will consider the values of \[x\] as \[x=m\] and \[x=n\]. Now, considering the product \[(x-m)(x-n)\] we will get the required factored form.