Question

How do you factor $49{{x}^{2}}-144?$

Answer 1

Hint: Remember the theorem we have learnt already: the product of sum and difference of two values is equal to the difference of squares of the values. Algebraically, $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}.$

Complete step by step solution:
Let us consider the given polynomial $49{{x}^{2}}-144$ which is to be factored.
To factorize this polynomial, we use the theorem that equates the product of sum and difference of two values and the difference of squares of those values.
If the two values mentioned in the above theorem are $a$ and $b,$ we can explain how this is true.
First, we are supposed to consider the product of sum and difference of $a$ and $b.$
That is, $\left( a+b \right)\left( a-b \right).$
Now let us do the multiplication of each term in the first parenthesis with each term of the second parenthesis.
$\Rightarrow \left( a+b \right)\left( a-b \right)={{a}^{2}}-ab+ab-{{b}^{2}}.$
Since $ab$ and $-ab$ are present in the above obtained equation, these two get cancelled, or we can say we cut them out.
Then we get,
$\Rightarrow \left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}.$
So, we have proved the theorem by obtaining the above equation.
Now, we are free to use this equation to factor the given expression into the product of its divisors.
Let us compare the variables of the given polynomial with the variables of the above derived identity.
Before that we can see, $49{{x}^{2}}={{\left( 7x \right)}^{2}}$ and $144={{12}^{2}}.$
Now we get,
$\Rightarrow 49{{x}^{2}}-144={{\left( 7x \right)}^{2}}-{{12}^{2}}.$
The comparison will give us, $a=7x$ and $b=12.$
Therefore, we get
$\Rightarrow 49{{x}^{2}}-144=\left( 7x+12 \right)\left( 7x-12 \right).$
Therefore, the factors are $7x+12$ and $7x-12.$

Note: We can also use grouping as follows:
The given polynomial is $49{{x}^{2}}-144.$
Add $84x$ and $-84x$ to this polynomial,
$\Rightarrow 49{{x}^{2}}-144=49{{x}^{2}}+84x-84x-144.$
Factor out $7x$ from the first two terms and $-12$ from the last two terms,
$\Rightarrow 49{{x}^{2}}-144=7x\left( 7x+12 \right)-12\left( 7x+12 \right).$
Factor out $\left( 7x+12 \right),$
$\Rightarrow 49{{x}^{2}}-144=\left( 7x-12 \right)\left( 7x+12 \right).$