Question

How do you factor the expression $7{{x}^{2}}-42x-49$?

Answer 1

Hint: First we will simplify the given equation by dividing the equation by 7 then we observe that the obtained equation is of the form $a{{x}^{2}}+bx+c=0$. Then the obtained equation is a quadratic equation so we will solve the given equation by using the quadratic formula method. The quadratic formula is given as $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.

Complete step by step solution:
We have been given an equation $7{{x}^{2}}-42x-49$.
We have to find the factors of the given equation.
We know that the general quadratic equation is of the form $a{{x}^{2}}+bx+c=0$. Then by dividing the given equation by 7 we will get
$\Rightarrow \dfrac{7{{x}^{2}}}{7}-\dfrac{42x}{7}-\dfrac{49}{7}=0$
Now, simplifying the above obtained equation we will get
$\Rightarrow {{x}^{2}}-6x-7=0$
Now, by comparing the obtained equation with the general equation we will get the values as
$a=1,b=-6,c=-7$
Now, the quadratic formula to solve the quadratic equation is given as $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Now, substituting the values we will get
$\Rightarrow x=\dfrac{-\left( -6 \right)\pm \sqrt{{{\left( -6 \right)}^{2}}-4\times 1\times -7}}{2\times 1}$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow x=\dfrac{6\pm \sqrt{36+28}}{2} \\
 & \Rightarrow x=\dfrac{6\pm \sqrt{64}}{2} \\
 & \Rightarrow x=\dfrac{6\pm 8}{2} \\
\end{align}$
Now, we have to consider both signs one by one then we will get
$\Rightarrow x=\dfrac{6+8}{2},x=\dfrac{6-8}{2}$
Now, simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow x=\dfrac{14}{2},x=\dfrac{-2}{2} \\
 & \Rightarrow x=7,x=-1 \\
\end{align}$
Hence we get the two factors of the given expression as $\left( x-7 \right)$ and $\left( x+1 \right)$.

Note: The given quadratic equation is of order two so it has two solutions. The number of solutions of an equation depends on its order or degree. The order of the equation is defined by the highest power of the variable the equation has. Also we have to consider both signs one by one to obtain the solution. Students may consider the solution of the equation as $x=7,x=-1$ but here in this question we have to factor the expression so we have to write the answer in factor form.