Question

How do you factorize ${{x}^{2}}+7x+6=0$?

Answer 1

Hint: Now to factorize the given quadratic equation we will first find the roots of the quadratic equation. Now we know that the roots of the equation of the form $a{{x}^{2}}+bx+c=0$ is given by the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Now we know that if $\alpha $ is the root of the equation then $\left( x-\alpha \right)$ is the factor of equation. Hence we will find the roots of the equation and then find the factors of the equation.

Complete step by step solution:
Now the given equation is a quadratic equation of the form $a{{x}^{2}}+bx+c=0$ .
Ow we know that there are two roots of the given equation.
To find the factors of the equation we will first find the roots of the equation.
Now let $\alpha $ and $\beta $ be the roots of the equation. Then we know that $\left( x-\alpha \right)$ and $\left( x-\beta \right)$ are the factors of the equation.
Now again we know that the roots of the quadratic equation of the form $a{{x}^{2}}+bx+c=0$ is given by $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .
Now consider the given equation ${{x}^{2}}+7x+6=0$ .
By comparing the equation with $a{{x}^{2}}+bx+c=0$ we get, a = 1, b = 7 and x = 6.
Hence the roots of the equation are,
$\begin{align}
  & \Rightarrow x=\dfrac{-7\pm \sqrt{{{7}^{2}}-4\left( 1 \right)\left( 6 \right)}}{2\left( 1 \right)} \\
 & \Rightarrow x=\dfrac{-7\pm \sqrt{49-24}}{2} \\
 & \Rightarrow x=\dfrac{-7\pm \sqrt{25}}{2} \\
 & \Rightarrow x=\dfrac{-7+25}{2} \\
\end{align}$
Hence we have $x=\dfrac{-7+25}{2}$ or $x=\dfrac{-7-25}{2}$
Hence $x=\dfrac{18}{2}=9$ or $x=\dfrac{-32}{2}=-16$
Hence the value of x = 9 or x = - 16.
Now since 9 and 16 are the roots of the equation $\left( x-9 \right)$ and $\left( x+16 \right)$ are the factors of the equation.

Note: Now note that there are various ways to find the roots of a quadratic equation we can use the method of completing squares too. In this method we first make the coefficient of ${{x}^{2}}$ as 1 and then add and subtract ${{\left( \dfrac{b}{2a} \right)}^{2}}$ to the equation to simplify and form a complete square. Hence we can easily solve the obtained equation. Also note that the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ is derived from the method of complete square.