Question

How do you factor \[{{x}^{2}}-2x-24\] ?

Answer 1

Hint: We can do it in different methods. But here we will use sum and product method or grouping method to factorize the equation. First we have to split our middle to satisfy the constraints we have in the method. Once we have splitted the middle term we have to rearrange the terms by taking common variables or numerical to arrive at our solution.

Complete step by step answer:
First we have to know the rules to split the middle term. They are
1. The sum of the two terms is equal to middle term and
2. The product must be equal to the product of first and last terms.
So we have to make sure to satisfy these conditions while splitting the middle term.
Given equation is
\[{{x}^{2}}-2x-24\]
Here to split the middle term the easy way to find the factors is to do prime factorization of the product of the first and last terms.
So in our case we have found the prime factors for \[24\].
The prime factors for \[24\] are
\[24=2\times 2\times 2\times 3\]
From the prime factors we have to take two factors through which we will get \[-2\] and \[-24\].
We can see that
\[-2=-6+4\]
\[-24=-6\times 4\]
So we can split our middle term into \[-6\] and \[4\]
Then the equation will look like
\[\Rightarrow {{x}^{2}}-6x+4x-24\]
Now we can rearrange the terms by taking common terms
From the above equation we can take \[x\] as common in first two terms and \[4\] as common in next two terms
Then the equation will become
\[\Rightarrow x\left( x-6 \right)+4\left( x-6 \right)\]
Now we can take \[x-6\] as common in two terms we will get
\[\Rightarrow \left( x-6 \right)\left( x-4 \right)\]

So the factors of the equation \[{{x}^{2}}-2x-24\] are \[\left( x-6 \right)\left( x-4 \right)\].

Note: We can also solve the problem using quadratic formulas also. But if the coefficient of \[{{x}^{2}}\] is \[1\] then it is better to use a grouping method. It will help you to solve the problem easily. But in this case you have to be sure while splitting up the middle term whether it is positive or negative.