Question

How do you factor $ 15{x^2} + 6x $ ?

Answer 1

Hint: In order to solve the given question we will first try to factorize them separately. After factorizing them on their own, we will then try to find which numbers are the common factors of both the numbers. In this case, 3 and $ x $ will be the common factors of both $ 15{x^2} $ and $ 6x $ .Therefore after taking it out the common from both the terms we will get a modified version of $ 15{x^2} + 6x $ in the form of $ x \times y $ where $ x $ will be $ 3x $ and $ y $ will be $ 5x + 2 $ .

Complete step-by-step answer:
The question has given us $ 15{x^2} + 6x $
So, in order to start factorizing it, we will need to factorize each component of the equation. So,
 $ 15{x^2} = 5 \times 3 \times x \times x....................(1) $
And
 $ 6x = 3 \times 2 \times x....................(2) $
If we compare (1) and (2) we can see that, both the numbers have a 3 and x in common
So, we can now rewrite the original form as:-
 $ 15{x^2} + 6x = (3 \times 5 \times x \times x) + (3 \times 2 \times x) $
Taking the common terms outside, our equation will become
 $ (3 \times 5 \times x \times x) + (3 \times 2 \times x) = (3x)\left[ {5x + 2} \right] $
We have successfully factorized the question $ 15{x^2} + 6x $ into two factors. One is $ 3x $ and the other is $ 5x + 2 $ .
Now, to verify whether we have found the correct factors or not, just open the brackets and multiply each term of $ 5x + 2 $ with $ 3x $
So,
 \[3x\left[ {5x + 2} \right] = (3x \times 5x) + (3x \times 2) \\
   \Rightarrow 15x + 6x \;
\]
We saw that when we open the brackets, our initial question forms up which signifies that our factors are correct and $ 3x $ and $ 5x + 2 $ are indeed the factors of $ 15{x^2} + 6x $ .
So, the correct answer is “ $ 3x $ and $ 5x + 2 $ ”.

Note: While calculating the common factors of both the terms, it is necessary that you must be careful while re writing the equation, as the modified form must be equal to the initial form when you're multiplying the terms again. Also, the common term that we take out from the given question is called as the GCF (Greatest Common Factor) of two numbers.