Question

How do you factor: - \[{{x}^{2}}+3x-54\]?

Answer 1

Hint: Use the middle term split method to factorize \[{{x}^{2}}+3x-54\]. Split 3x into two terms in such a way that their sum is 3x and the product is \[-54{{x}^{2}}\]. For this process, find the prime factors of 54 and combine them in such a manner so that we can get our condition satisfied. Finally, take the common terms together and write \[{{x}^{2}}+3x-54\] 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 solution:
Here, we have been asked to factorize the quadratic polynomial: \[{{x}^{2}}+3x-54\].
Let us use the middle term split method for the factorization. It says that we have to split the middle term which is 3x into two terms such that their sum is 3x and product is \[-54{{x}^{2}}\]. To do this, first we need to find all the prime factors of 54. So, let us find.
We know that 54 can be written as: - \[\text{54}=2\times 3\times 3\times 3\] as the product of its primes. Now, we have to group these factors such that the conditions of the middle term split method are satisfied. So, we have,
(i) $\left( 9x \right)+\left( -6x \right)=3x$
(ii) \[\left( 9x \right)\times \left( -6x \right)=-54{{x}^{2}}\]
Hence, both the conditions of the middle term split method are satisfied. So, the quadratic polynomial can be written as: -
\[\begin{align}
  & \Rightarrow {{x}^{2}}+3x-54={{x}^{2}}+9x-6x-54 \\
 & \Rightarrow {{x}^{2}}+3x-54=x\left( x+9 \right)-6\left( x+9 \right) \\
\end{align}\]
Grouping the common terms together we have,
\[\Rightarrow {{x}^{2}}+3x-54=\left( x+9 \right)\left( x-6 \right)\]
Hence, \[\left( x+9 \right)\left( x-6 \right)\] 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: $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ to get the two values of x. Now, we will consider the values of x as x = m n x = n. Now, considering the product (x – m) (x – n) we will get the required factored form.