Question

How do you factorize the trinomial ${x^2} + 8x - 20$ ?

Answer 1

Hint: In this question, we are given a quadratic equation and we have been asked 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 ${x^2} + 8x - 20$ .
Let us find two such factors of $ - 20$, such that when they are added or subtracted, they give us $8$.
If we observe, then two such factors are $10$ and $2$. But, if we add them, we will get $12$ and we want $8$. So, we will take one factor in negative. Our required factors are $10$ and $ - 2$.
$ \Rightarrow {x^2} + 10x - 2x - 20$
Taking $x$ common from the first two terms and $2$ from the last two terms,
$ \Rightarrow x\left( {x + 10} \right) - 2\left( {x + 10} \right)$
Making factors,
$ \Rightarrow \left( {x + 10} \right)\left( {x - 2} \right)$

Hence, $(x+10)$ and $(x-2)$ are the factors of ${x^2} + 8x - 20$.

Note: 1) What is a Trinomial?
A trinomial is a mathematics equation, having three terms, connected by the signs of addition or subtraction. For example: ${x^2} + 8x - 20$ has three terms ${x^2}$, $8x$ and $20$.
2) How to decide which factor will be negative?
We had two factors - $10$ and $2$. On what basis did we take $2$ as a negative and $10$ as a non-negative factor?
Our equation ${x^2} + 8x - 20$ has a middle term as positive $8x$. If we had taken $10$ as negative, our term would have been $ - 8x$. Hence, we choose the factor to be negative as per the sign of the term in the given equation.