Question

How do you factor completely: $3{x^2} - 21x + 30$?

Answer 1

Hint:
First of all, we will take 3 common out of all the terms and then, we will use the method of splitting the middle term and then combine the first two and then the last two together to get the required answer.

Complete step by step solution:
We are given that we are required to factor completely the quadratic equation: $3{x^2} - 21x + 30$.
Now, we see that 3, 21 and 30, all of these numbers have 3 common in them. Therefore, we will take out 3 common from all of these and obtain the following expression:-
$ \Rightarrow 3{x^2} - 21x + 30 = 3\left( {{x^2} - 7x + 10} \right)$
Now, we will break the middle term and then, we can write it as the following expression:-
$ \Rightarrow 3\left( {{x^2} - 2x - 5x + 10} \right)$
Now, we can take x common out of the first two terms in the bracket, then we will obtain the following expression with us:-
$ \Rightarrow 3\left\{ {x\left( {x - 2} \right) - 5x + 10} \right\}$
As we did in the last step, we can also take out 5 common from the last two terms in the bracket above, we will then obtain the following expression with us:-
$ \Rightarrow 3\left\{ {x\left( {x - 2} \right) - 5\left( {x + 2} \right)} \right\}$
Now, since we have (x + 2) in both the terms in the above expression, we can take it out as well to obtain the following expression with us:-
$ \Rightarrow 3\left( {x - 2} \right)\left( {x - 5} \right)$
Thus, we have the required answer.

Note:
The students must note that there is an alternate way to solve the same question.
Alternate way:
$ \Rightarrow 3{x^2} - 21x + 30 = 3\left( {{x^2} - 7x + 10} \right)$
Now, we will find the roots of the equation ${x^2} - 7x + 10 = 0$.
Now, we know that if we have an equation given by $a{x^2} + bx + c = 0$ , then its roots are given by the following expression:-
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Putting a = 1, b = - 7 and c = 10, we will then obtain the following expression:-
$ \Rightarrow x = \dfrac{{ - ( - 7) \pm \sqrt {{{( - 7)}^2} - 4(1)(10)} }}{{2(1)}}$
Simplifying the squares and multiplications in the above expression, we will obtain:-
$ \Rightarrow x = \dfrac{{7 \pm \sqrt {49 - 40} }}{2}$
Solving it, we will then obtain:-
$ \Rightarrow x = 5,2$
Thus, we can write the given expression as $3\left( {x - 2} \right)\left( {x - 5} \right)$.