How do you find the factor $3{x^2} - 8x + 15?$

Hint: According to the question we have to determine the factor of the given expression which is $3{x^2} - 8x + 15$. So, first of all we have to comparing the given expression $3{x^2} - 8x + 15$ with the standard form of the equation that is $a{x^2} + bx + c$ and find the value of $a,b$ and $c$,where $a$ is the coefficient of ${x^2}$, $b$ is the coefficient of $x$ and $c$ is the constant term of the given expression.
Now, we have to factor the given quadratic equation as $3{x^2} - 8x + 15$with the help of the formula as mentioned below.

Formula used: $ \Rightarrow \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}..................................(A)$

Complete step-by-step solution:
Step 1: So, first of all we have to comparing the given expression $3{x^2} - 8x + 15$ with the standard form of the equation that is $a{x^2} + bx + c$ and find the value of $a,b$ and $c$.
$ \Rightarrow a = 3,b = - 8,c = 15$
Step 2: Now, we have to factor the given quadratic equation as $3{x^2} - 8x + 15$ with the help of the formula (A) as mentioned in the solution hint.
$
\Rightarrow \dfrac{{ - \left( { - 8} \right) \pm \sqrt {{{\left( { - 8} \right)}^2} - 4\left( 3 \right)\left( {15} \right)} }}{{2\left( 3 \right)}} \\
 \Rightarrow \dfrac{{8 \pm \sqrt {64 - 180} }}{6} \\
 \Rightarrow \dfrac{{8 \pm \sqrt { - 116} }}{6}
 $
Step 3: Now, as we know that the value of $\sqrt { - 1} $ is $i$ and the value of $\sqrt {116} $ is $10.77$.So, put these values in the expression obtain in the solution step 2.
$ \Rightarrow \dfrac{{8 \pm 10.77i}}{6}$
So, the roots or factors are $\dfrac{{8 + 10.77i}}{6}$ and $\dfrac{{8 - 10.77i}}{6}$

Hence, the factors of the given quadratic equation as $3{x^2} - 8x + 15$ is $\dfrac{{8 + 10.77i}}{6}$ and $\dfrac{{8 - 10.77i}}{6}$

Note: It is necessary to compare the given expression as $3{x^2} - 8x + 15$ with the standard form of the equation that is $a{x^2} + bx + c$and find the value of $a,b$and$c$.
It is necessary to factor the given quadratic equation as $3{x^2} - 8x + 15$with the help of the formula (A) as mentioned in the solution hint.