Question

How do you factor $6{r^2} - 28r + 16$?

Answer 1

Hint: This equation is the quadratic equation. The general form of the quadratic equation is $a{x^2} + bx + c = 0$. Where ‘a’ is the coefficient of ${x^2}$, ‘b’ is the coefficient of x and ‘c’ is the constant term.
To solve this equation, we will apply the sum-product pattern. During the simplification, we will take out common factors from the two pairs. Then we will rewrite it in factored form.
Therefore, we should follow the below steps:
Apply sum-product patterns.
Make two pairs.
Common factor from two pairs.
Rewrite in factored form.

Complete step-by-step answer:
 Here, the quadratic equation is
$ \Rightarrow 6{r^2} - 28r + 16$
Let us take out the common factor 2 from the equation.
$ \Rightarrow 2\left( {3{r^2} - 14r + 8} \right)$
Let us apply the sum-product pattern in the above equation.
Since the coefficient of ${x^2}$is 3 and the constant term is 8. Let us multiply 3 and 8. The answer will be 24. We have to find the factors of 24 which sum to -14. Here, the factors are -12 and -2.
Therefore,
$ \Rightarrow 2\left( {3{r^2} - 12r - 2r + 8} \right)$
Now, make two pairs in the above equation.
$ \Rightarrow 2\left[ {\left( {3{r^2} - 12r} \right) - \left( {2r - 8} \right)} \right]$
Let us take out the common factor.
$ \Rightarrow 2\left[ {3r\left( {r - 4} \right) - 2\left( {r - 4} \right)} \right]$
Now, rewrite the above equation in factored form.
$ \Rightarrow 2\left( {3r - 2} \right)\left( {r - 4} \right)$
Hence, the factors of the given equation are 2, $\left( {3r - 2} \right)$, and $\left( {r - 4} \right)$.

Note:
One important thing is, we can always check our work by multiplying our factors back together, and check that we have got back the original answer.
To check our factorization, multiplication goes like this:
$ \Rightarrow 2\left( {3r - 2} \right)\left( {r - 4} \right)$
Let us apply multiplication to remove brackets.
$ \Rightarrow 2\left( {3{r^2} - 12r - 2r + 8} \right)$
Let us simplify it. We will get,
$ \Rightarrow 2\left( {3{r^2} - 14r + 8} \right)$
That is equal to,
$ \Rightarrow 6{r^2} - 28r + 16$
Hence, we get our quadratic equation back by applying multiplication.
Here is a list of methods to solve quadratic equations:
Factorization
Completing the square
Using graph
Quadratic formula