Question

How do you factor $3{{x}^{2}}-3x-8$

Answer 1

Hint: Now we are given with a quadratic expression of the form $a{{x}^{2}}+bx+c$ . Now we will use the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ to find the roots of the quadratic expression. Now once we have the roots of the expression we know that the factors are given by $x-\alpha $ and $x-\beta $ where $\alpha $ and $\beta $ are the roots of the expression. Hence we have the factors of the given expression.

Complete step by step solution:
Now we know that the given expression is a quadratic expression in x.
Now first we will find the roots of the given expression.
Now comparing the given expression $3{{x}^{2}}-3x-8$ with the general form of quadratic expression $a{{x}^{2}}+bx+c$ we get a = 3, b = -3 and c = -8.
Now we know that the roots of the expression $a{{x}^{2}}+bx+c$ is given by the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .
Hence we get the roots of the expression is $x=\dfrac{-\left( -3 \right)\pm \sqrt{{{\left( -3 \right)}^{2}}-4\left( 3 \right)\left( -8 \right)}}{2\left( 3 \right)}$
$\begin{align}
  & \Rightarrow x=\dfrac{3\pm \sqrt{9+96}}{6} \\
 & \Rightarrow x=\dfrac{3\pm \sqrt{105}}{6} \\
\end{align}$
Now we know that the roots of the given expression are $\dfrac{3+\sqrt{105}}{6}$ and $\dfrac{3-\sqrt{105}}{6}$ .
Now we know that if $\alpha $ and $\beta $ are the roots of the expression then $x-\alpha $ and $x-\beta $ are the factors of the expression.
Hence we get the factors of the expression are \[x-\left( \dfrac{3+\sqrt{105}}{6} \right)\] and \[x-\left( \dfrac{3-\sqrt{105}}{6} \right)\].
Hence the given expression is factored

Note: Now note that here we have used the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ to find the roots of the expression. We can also find the roots by using the complete square method. To find the roots by this method we will first divide the whole expression by coefficient of ${{x}^{2}}$ . Now we will add and subtract the expression of the form $a{{x}^{2}}+bx+c$ by ${{\left( \dfrac{b}{2a} \right)}^{2}}$ . Now we will further simplify the expression by $\left( {{a}^{2}}\pm {{b}^{2}} \right)={{a}^{2}}\pm 2ab+{{b}^{2}}$ . Now we will simplify the expression and hence take square root on both sides to eliminate the powers and find the roots of the expression.