Question

How do you factor ${x^2} + 25x + 150$ ?

Answer 1

Hint: In this question, we are given a quadratic expression and we have to factorize it. Use splitting the middle term method to find the factors. Let us assume that the equation is in the form of $a{x^2} + bx + c$. If ${x^2}$ does not have a coefficient, then find two factors of $c$, such that when they are added or subtracted, they give us $b$. But, if ${x^2}$ has a coefficient, then find two factors of $ac$, such that when they are added or subtracted, they give us $b$. After finding factors, take out the common numbers or variables and factorize the expression.

Complete step-by-step solution:
We are given a quadratic expression ${x^2} + 25x + 150$.
Since our ${x^2}$ does not have any coefficient, we will find the factors of $c = 150$.
Let us find two such factors of $150$, such that when they are added or subtracted, they give us $25$.
If we observe, then two such factors are $30$ and $5$.
In order to get $25$, one of the factors needs to be negative. But, if one factor is negative, we will get $ - 150$, and we want $150$.
So, another pair of factors can be $15$ and $10$. Our required factors are $15$ and $10$.
$ \Rightarrow {x^2} + 15x + 10x + 150$
Taking $x$ common from the first two terms and $10$ from the last two terms,
$ \Rightarrow x\left( {x + 15} \right) + 10\left( {x + 15} \right)$
Making factors,
$ \Rightarrow \left( {x + 15} \right)\left( {x + 10} \right)$

Hence, $(x+15)$ and $(x+10)$ are the factors of ${x^2} + 25x + 150$.

Note: Using these factors, we can also find the values of $x$.
We just have to put each factor equal to $0$ and then, we have to shift the terms to the other side. We will get the required values as well.
$ \Rightarrow x + 15 = 0,x + 10 = 0$
Shifting terms to the other side,
$ \Rightarrow x = - 15, - 10$
Hence, these are the required values of x.