Question

How do you factor the expression $ - 4{x^2} + 10x + 24$?

Answer 1

Hint: In order to factor the expression, we first take out the common number from the expression, then we use a grouping method to factorize the remaining equation. We split the middle term into two terms according to the sum- product rule, which states that the middle term should be split in such a way that the two terms are product of the first and the last numbers, and the sum of the terms gives us the original middle term.

Complete step-by-step solution:
Given expression is $ - 4{x^2} + 10x + 24$
Let us take out the common factor from the above given equation:
$ \Rightarrow - 4{x^2} + 10x + 24$, here $ - 2$ is a common factor
Thus, the equation becomes: $ - 2\left( {2{x^2} - 5x - 12} \right)$
Now, let’s solve the equation within the brackets:
 $ \Rightarrow - 2\left( {2{x^2} - 5x - 12} \right)$………. Equation (A)
Here the middle term is $ - 5x$ , we need to split the middle term into two terms in such a way so that the product of the terms is equal to the product of the first term $\left( 2 \right)$ and third term $\left( { - 12} \right)$ and the sum of the two terms give us the original middle term.
The product of the first term and third terms is $ - 24$ , in order to split the middle term, we find the factors of the $ - 24$
Factors of $ - 24$ which when added together will give us the middle term $5x$ is $8,3$
Thus, we split the middle term: $ - 5x = - 8x + 3x$
Placing the above mentioned middle term in equation (A), we get:
$ \Rightarrow - 2\left( {2{x^2} - 8x + 3x - 12} \right)$
Now we need to make two groups and find common factors:
$ \Rightarrow - 2\left( {2x\left( {x - 4} \right) + 3\left( {x - 4} \right)} \right)$
Here $\left( {x - 4} \right)$ is a common factor
$ \Rightarrow - 2\left( {\left( {2x + 3} \right)\left( {x - 4} \right)} \right)$

Hence the factor of the given expression is $ - 2\left( {\left( {2x + 3} \right)\left( {x - 4} \right)} \right)$.

Note: Factorization is simply the method of breaking down a given expression into their simplest factors. Grouping method used to solve the sum above is the most common method of solving any quadratic equation. One should also be attentive towards the signs placed in front of the numbers, as many students make a mistake in getting confused with the signs.