Question

How do you factor the expression $4{{b}^{2}}-9?$

Answer 1

Hint: To factor the expression $4{{b}^{2}}-9,$ or in other words we can say that to find the factors of the expression, we use the following theorem: The difference ${{a}^{2}}-{{b}^{2}}$ of two perfect squares can be factored into the product of $\left( a+b \right)$ and $\left( a-b \right).$ That can be written in the mathematical form as, ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right).$

Complete step by step solution:
We are given with the expression $4{{b}^{2}}-9.$
We have to factor it to get all the factors it has.
Now, we know the theorem that says that the product of sum and difference of two values is equal to the difference of the squares of those values.
Suppose that $x$ and $y$ are the two values mentioned in the theorem that is written above.
Then, the following mathematical approach will prove this theorem and then we will be directed to the factors of the given expression.
Consider the product, $\left( x+y \right)\left( x-y \right).$
Now we are going to expand this product.
So, we get,
$\Rightarrow \left( x+y \right)\left( x-y \right)={{x}^{2}}-xy+xy-{{y}^{2}}$
The same terms with opposite signs get cancelled to give,
$\Rightarrow \left( x+y \right)\left( x-y \right)={{x}^{2}}-{{y}^{2}}.$
Therefore, it is proven that,
\[\Rightarrow {{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right).\]
Again, consider the expression we want to factor,
$\Rightarrow 4{{b}^{2}}-9.$
We know that $4{{b}^{2}}={{\left( 2b \right)}^{2}}$ and $9={{3}^{2}}.$
Now we can write,
$\Rightarrow 4{{b}^{2}}-9={{\left( 2b \right)}^{2}}-{{3}^{2}}.$
Thus, we get $x=2b$ and $y=3.$
And from the above proof, we have \[{{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right).\]
Apply this theorem in the given expression to be factored,
$\Rightarrow 4{{b}^{2}}-9={{\left( 2b \right)}^{2}}-{{3}^{2}}=\left( 2b-3 \right)\left( 2b+3 \right).$
Therefore, the factors are $\left( 2b-3 \right)$ and $\left( 2b+3 \right).$

Note: There is an alternative method for factoring the given expression:
Take the given expression, $4{{b}^{2}}-9.$
Let us add $6b$ and $-6b$ to this expression, we get
$\Rightarrow 4{{b}^{2}}-9=4{{b}^{2}}+6b-6b-9.$
Take the common factors outside, then we will get,
$\Rightarrow 4{{b}^{2}}-9=2b\left( 2b+3 \right)-3\left( 2b+3 \right).$
Again, we are taking the common factor out.
Then we get,
\[\Rightarrow 4{{b}^{2}}-9=\left( 2b-3 \right)\left( 2b+3 \right).\]
Therefore, the factors of the expression \[4{{b}^{2}}-9\] are $2b-3$ and $2b+3.$
The factorization is \[4{{b}^{2}}-9=\left( 2b-3 \right)\left( 2b+3 \right).\]