Question

How do you factor $4{{m}^{2}}-25?$

Answer 1

Hint: We factorize the expression $4{{m}^{2}}-25$ to find all the factors it has. We do it by grouping. First, we add and subtract a term to and from the expression. Then we group the terms in the obtained expression.

Complete step-by-step solution:
Let us take the given expression into consideration.
That is, $4{{m}^{2}}-25.$
Now we are going to add a $10m$ to the expression while we subtract a $10m$ from the expression,
$\Rightarrow 4{{m}^{2}}-25=4{{m}^{2}}+10m-10m-25.$
Consider the first two terms, namely $4{{m}^{2}}$ and $10m.$ The term $2m$ is common in both the terms. Let us take that out. Then, $4{{m}^{2}}+10m=2m\left( 2m+5 \right).$
Similarly, consider the last two terms in the expression, namely $-10m$ and $25.$ The term $-5$ is common in both of them. Take that out, $-10m-25=-5\left( 2m+5 \right).$
So, we will get,
$\Rightarrow 4{{m}^{2}}-25=2m\left( 2m+5 \right)-5\left( 2m+5 \right).$
In the above obtained expression, the term $2m+5$ is common. So, we are taking it out.
Then we get,
\[\Rightarrow 4{{m}^{2}}-25=\left( 2m-5 \right)\left( 2m+5 \right).\]
Therefore, the factors of the expression \[4{{m}^{2}}-25\] are $2m-5$ and $2m+5.$
The factorization is \[4{{m}^{2}}-25=\left( 2m-5 \right)\left( 2m+5 \right).\]

Note: This can be done easily by using another method. This method includes the following theorem: A difference of two perfect squares ${{x}^{2}}$ and ${{y}^{2}}$ can be factored into a product of the sum (of $x$ and $y$) and the difference (of $x$ and $y$).
That can be written, algebraically, as ${{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right).$
Consider the given expression, $4{{m}^{2}}-25.$
Here, ${{x}^{2}}=4{{m}^{2}}$ and ${{y}^{2}}=25.$
From this we will get ${{x}^{2}}={{\left( 2m \right)}^{2}}$ and $y={{5}^{2}}.$
And, so, we have $x=\left( 2m \right)$ and $y=5.$
From what we have found we can easily find the factors and factorization of the given expression using the above given theorem.
Thus, when we apply the theorem, we get the following equation which is the factorization of the given expression.
So, we get
$\Rightarrow 4{{m}^{2}}-25=\left( 2m+5 \right)\left( 2m-5 \right).$