Question

How do you factor completely $3{x^2} + 4x - 15$?

Answer 1

Hint: This problem deals with factoring the given expression. This can be done either by the method of completing the square or just factoring and solving the quadratic equation. To solve $a{x^2} + bx + c = 0$, expression in $x$, by completing the square: transform the equation so that the constant term,$c$ is alone on the right side. But here we are adding and subtracting some terms in order to factor.

Complete step-by-step solution:
Given the quadratic expression is $3{x^2} + 4x - 15$, consider it as given below:
$ \Rightarrow 3{x^2} + 4x - 15$
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 45, now the factors of 45, which the difference of the factors is equal to the coefficient of the $x$ term are 9 and 5, where these product is also equal to 45, hence the term $4x$ is expressed as the difference of $9x$ and $5x$, as shown below:
$ \Rightarrow 3{x^2} + 9x - 5x - 15$
Now taking the term $3x$ common from the first two terms, and taking the number 5 common from the second two terms, which is shown below:
$ \Rightarrow 3{x^2} + 9x - 5x - 15$
$ \Rightarrow 3x\left( {x + 3} \right) - 5\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 - 5} \right)$
So here we factorized the given quadratic expression into two factors, which is shown below:
$ \Rightarrow 3{x^2} + 4x - 15 = \left( {x + 3} \right)\left( {3x - 5} \right)$
One of the factors of $3{x^2} + 4x - 15$ is \[\left( {x + 3} \right)\] and the other factor is \[\left( {3x - 5} \right)\].

The factors of $3{x^2} + 4x - 15$ are \[\left( {x + 3} \right)\] and \[\left( {3x - 5} \right)\].

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 solve the value of $x$ from the given equation is directly equated to zero, as $3{x^2} + 4x - 15 = 0$, then after finding the roots, and then factorize.