Question

How do you factor completely $6{{x}^{2}}+7x+2$?

Answer 1

Hint: Now first we will find the roots of the quadratic equation. We know that the roots of quadratic of the form $a{{x}^{2}}+bx+c$ is given by $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Hence using this we will find the roots of the expression. Now if $\alpha $ and $\beta $ are the roots of the expression then $\left( x-\alpha \right)$ and $\left( x-\beta \right)$ are the factors of the expression. Hence we have factors of the given equation.

Complete step by step solution:
Now the given expression is a quadratic expression in x.
To factor the expression we will first find the roots of the given expression.
Now the roots are nothing but the values of x satisfying $6{{x}^{2}}+7x+2=0$ .
Now the general form of a quadratic equation is $a{{x}^{2}}+bx+c$ .
Hence comparing the given expression with this we get, a = 6, b = 7 and c = 2.
Now we know that the formula for quadratic equation of the form $a{{x}^{2}}+bx+c$ is given by $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .
Now we will substitute the values of a, b and c in the above formula to find the roots.
Hence we have the roots of the expression as $\Rightarrow x=\dfrac{-7\pm \sqrt{{{7}^{2}}-4\left( 6 \right)\left( 2 \right)}}{2a}$
Now on simplifying the above equation we get,
$\begin{align}
  & \Rightarrow x=\dfrac{-7\pm \sqrt{49-48}}{2\left( 6 \right)} \\
 & \Rightarrow x=\dfrac{-7\pm 1}{12} \\
\end{align}$
Hence we get $x=\dfrac{-1}{2}$ and $x=\dfrac{-2}{3}$
Hence we have the roots of the equation are $x=\dfrac{-1}{2}$ and $x=\dfrac{-2}{3}$
Now we know that if $\alpha $ is the root of the equation then $x-\alpha $ is the factor of the equation.
Hence we get the factors of the expression are $\left( x+\dfrac{1}{2} \right)\left( x+\dfrac{2}{3} \right)$ .
Hence we get, $6{{x}^{2}}+7x+2=\left( x+\dfrac{1}{2} \right)\left( x+\dfrac{2}{3} \right)$ .

Note: Now note that we can also use the complete square method to factor the expression.
To solve by this method we first divide the whole quadratic by a and then add and subtract ${{\left( \dfrac{b}{2a} \right)}^{2}}$ . Now we will simplify the obtained equation by using ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ . Hence we get a complete square in the equation. Now we will take square root such that we get linear equations in x and hence solve the equation to find roots.