Question

How do you factor $4{x^2} - 16x + 15$?

Answer 1

Follow these steps to achieve the solution:
First step is to split the middle term of the given polynomial equation.
Then use the distributive property.
After that take out the common factors.
And factorize generally to achieve the solution.

Complete Step by Step Solution:
Let us start by writing the given equation:
$\Rightarrow 4{x^2} - 16x + 15$
Here, we can see that the given equation is a polynomial equation of the form $a{x^2} + bx + c$.
Where, the values of the variables are as: $a = 4$ , $b = - 16$ and $c = 15$.
Rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot 15 = 60$ and whose sum is $b = - 16$.
First, write the given equation:
$\Rightarrow 4{x^2} - 16x + 15$
Now, factor 16 out of $16x$ we will get:
$\Rightarrow 4{x^2} - 16\left( x \right) + 15$
Then split the middle term like this:
$\Rightarrow 4{x^2} - \left( {6 + 10} \right)x + 15$
Now, to solve the above equation apply the distributive property and we will get:
$\Rightarrow 4{x^2} - 6x - 10x + 15$
Group the first two terms and the last two terms:
$\Rightarrow \left( {4{x^2} - 6x} \right) - \left( {10x - 15} \right)$
After this, factor out the greatest common factor from each of the group:
$\Rightarrow 2x\left( {2x - 3} \right) - 5\left( {2x - 3} \right)$
Factor the polynomial by factoring out the greatest common factor $\left( {2x - 3} \right)$.
$\Rightarrow \left( {2x - 3} \right)\left( {2x - 5} \right)$

Hence, the factor of the given equation $4{x^2} - 16x + 15$ is $\left( {2x - 3} \right)\left( {2x - 5} \right)$.

Note: Note it does not matter if you reverse the order of splitting $16x$ as $- 6x - 10x$ or $- 10x - 6x$, it is going to give the same result.
Remember that this type of question always uses distributive property in most of the cases.
You should also be familiar with the term (GCF) which stands for greatest common factor means the largest number that divides two or more numbers evenly.