Question

How do you factor $3{x^2} + 24$ ?

Answer 1

Hint: This problem deals with factoring the given expression in $x$. This can be done either by the method of completing the square or just factoring and solving the quadratic equation. To solve $a{x^2} + bx + c = 0$ 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 $3{x^2} + 24$, consider it as given below:
$ \Rightarrow 3{x^2} + 24$
We can observe that it is an expression with presence of ${x^2}$ term and a constant, where the $x$ term is missing in the given expression.
Here taking the term $3$ common from the first two terms, which is shown below:
$ \Rightarrow 3{x^2} + 24$
$ \Rightarrow 3\left( {{x^2} + 8} \right)$
So we now split the given quadratic expression into a product of two expressions, where one of the expressions is a zero polynomial or a number and the other expression is a polynomial of degree two.
So here we factorized the given quadratic expression into two factors.
$ \Rightarrow 3{x^2} + 24 = 3\left( {{x^2} + 8} \right)$
One factor is 3, and the other factor is $\left( {{x^2} + 8} \right)$

The factors of $3{x^2} + 24$ are $3$ and $\left( {{x^2} + 8} \right)$.

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 $64{x^2} - 1 = 0$, this can be done by sending the constant 1 to the right hand side of the equation and then solve for $x$, and then factorize with the obtained solutions.