Question

How do you factor the quadratic polynomial $4{x^2} - 24x + 32$?

Answer 1

Hint: First of all, take out the common factor of 4 from all the terms and then split the middle term to write in such a format that we get one common factor that is the method of splitting the middle term to get the factors.

Complete step-by-step solution:
We are given that we are required to factor $4{x^2} - 24x + 32$.
Let us say $f(x) = 4{x^2} - 24x + 32$.
We will first take out 4 common from the equation.
We can write the given equation as follows:-
$ \Rightarrow 4{x^2} - 24x + 32 = 4\left( {{x^2} - 6x + 8} \right)$
We can write the above mentioned equation as follows:-
$ \Rightarrow 4{x^2} - 24x + 32 = 4\left( {{x^2} - 2x - 4x + 8} \right)$
Taking x common from the first two terms in the latter factor, we will then obtain the following equation with us:-
$ \Rightarrow 4{x^2} - 24x + 32 = 4\left\{ {x\left( {x - 2} \right) - 4x + 8} \right\}$
Taking – 4 common from the last two terms in the latter factor, we will then obtain the following equation with us:-
$ \Rightarrow 4{x^2} - 24x + 32 = 4\left\{ {x\left( {x - 2} \right) - 4\left( {x - 2} \right)} \right\}$
Taking (x – 2) common from the last two terms in the latter factor, we will then obtain the following equation with us:-
$ \Rightarrow 4{x^2} - 24x + 32 = 4\left( {x - 2} \right)\left( {x - 4} \right)$
Thus, we have the required factors.

Note: The students must notice that we have an alternate way of factoring the quadratic equation involved in it as well. The alternate way is as follows:-
The given equation is ${x^2} - 6x + 8$.
Let us equate the given equation to 0 for once so that we can find its roots easily.
So, the equation becomes ${x^2} - 6x + 8 = 0$.
Using the quadratic formula given by if the equation is given by $a{x^2} + bx + c = 0$, its roots are given by the following equation:-
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Thus, we have the roots of ${x^2} - 6x + 8$ given by:
$ \Rightarrow x = \dfrac{{ - ( - 6) \pm \sqrt {{{( - 6)}^2} - 4 \times 8} }}{2}$
Simplifying the calculations in the square root in the numerator of the right hand side, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{{6 \pm \sqrt {36 - 32} }}{2}$
Simplifying the calculations in the square root in the numerator of the right hand side further, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{{6 \pm 2}}{2}$
Hence, the roots are 2 and 4.
$ \Rightarrow 4{x^2} - 24x + 32 = 4\left( {x - 2} \right)\left( {x - 4} \right)$
Thus, we have the required factors.