Question

How do you factor \[3{x^2} + 5x - 12\]?

Answer 1

Hint: This problem deals with factorizing the given expression and then finding the zeros of the given function. This can be done either by the method of completing the square or just factoring and solving the quadratic equation. To solve $a{x^2} + bx + c$, expression in $x$, by factoring the given expression. But here we are adding and subtracting some terms in order to factor.

Complete step-by-step solution:
Given the quadratic expression is \[3{x^2} + 5x - 12\], consider it as given below:
$ \Rightarrow 3{x^2} + 5x - 12$
Now expressing the above expression such that the $x$ term is split into the factors of the product of the ${x^2}$ term and the constant term which is equal to 36, now the factors of 36, which the difference of the factors is equal to the coefficient of the $x$ term are 9 and 4, where these product is also equal to 36, hence the term $5x$ is expressed as the difference of $9x$ and $4x$, as shown below:
$ \Rightarrow 3{x^2} + 5x - 12$
$ \Rightarrow 3{x^2} + 9x - 4x - 12$
Now taking the term $3x$ common from the first two terms, and taking the number -4 common from the second two terms, which is shown below:
$ \Rightarrow 3x\left( {x + 3} \right) - 4\left( {x + 3} \right)$
Now taking the term $\left( {x + 3} \right)$ common in the above expression, as shown below:
$ \Rightarrow \left( {x + 3} \right)\left( {3x - 4} \right)$
So here we factorized the given quadratic expression into two factors, which is shown below:
$ \Rightarrow 3{x^2} + 5x - 12 = \left( {x + 3} \right)\left( {3x - 4} \right)$

The factors of \[3{x^2} + 5x - 12\] are $\left( {x + 3} \right)$ and $\left( {3x - 4} \right)$.

Note: Please note that this problem can also be solved by another method, which is described here. Instead of first factoring and then solving for $x$, we can directly the value of $x$ from the given equation is directly equated to zero, as \[3{x^2} + 5x - 12 = 0\], then after finding the roots, and then factorize.