Question

How do you factor $3{x^2} + 5x + 2$ ?

Answer 1

Hint: In this question, we will use the concept of the middle term splitting for the quadratic equation. As we know that the factors of the quadratic equation can be calculated by splitting the middle term such that the sum of the numbers is equal to the coefficient of the $x$ while the product of the number should be equal to the coefficients of ${x^2}$ and the constant.

Complete step by step solution:
In this question, we have given a function $3{x^2} + 5x + 2$, we need to provide the factor of the given function.
In the given function, first we will split $5x$ as $\left( {3x + 2x} \right)$ and the expression become,
$ \Rightarrow h\left( x \right) = 3{x^2} + 3x + 2x + 2$
Now, we will form the combination of the terms as,
$ \Rightarrow h\left( x \right) = \left( {3{x^2} + 3x} \right) + \left( {2x + 2} \right)$
From the above expression we can see that we can take $3x$ as common from the first term and $2$ from the second term combination, so the expression become,
$ \Rightarrow h\left( x \right) = 3x\left( {x + 1} \right) + 2\left( {x + 1} \right)$
Now, we will take $\left( {x + 1} \right)$ as the common from each term and obtain the following factors,
\[\therefore h\left( x \right) = \left( {x + 1} \right)\left( {3x + 2} \right)\]

Therefore, the factors of the given expression are \[\left( {x + 1} \right)\left( {3x + 2} \right)\].

Note:
We can also obtain the factors by using formula. As we know that if the quadratic equation is in the form $a{x^2} + bx + c$, then the roots of that equation will be,
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Now, we will apply this formula for $3{x^2} + 5x + 2$ and obtain,
$ \Rightarrow {x_{1,2}} = \dfrac{{ - 5 \pm \sqrt {{{\left( 5 \right)}^2} - 4\left( 3 \right)\left( 2 \right)} }}{{2\left( 3 \right)}}$
Now, we will simplify the above expression and obtain the following roots of the given quadratic equation,
$ \Rightarrow {x_{1,2}} = \dfrac{{ - 5 \pm \sqrt {25 - 24} }}{6}$
Now, we will be simplified it further to obtain,
$ \Rightarrow {x_{1,2}} = \dfrac{{ - 5 \pm 1}}{6}$
After simplification we will get,
$ \Rightarrow {x_1} = - \dfrac{2}{3}$
$ \Rightarrow {x_2} = - 1$
From the above roots we can write it as,
$3{x^2} + 5x + 2 = \left( {3x + 2} \right)\left( {x + 1} \right)$
Therefore, the factors of the given expression are \[\left( {x + 1} \right)\left( {3x + 2} \right)\].