Question

How do you check that you factored a quadratic correctly?

Answer 1

Hint: Here in this question we have been asked to check if a quadratic equation is factored correctly or not. For doing that we will assume an equation and factorize it and then verify it. Let us assume the quadratic equation to be ${{x}^{2}}+4x+4=0$ .

Complete step by step solution:
Now considering from the question we have been asked to check if a quadratic equation is factored correctly or not.
For doing that we will assume an equation and factorize it and then verify it.
Let us assume the quadratic equation to be ${{x}^{2}}+4x+4=0$ .
From the basic concepts we know that the roots of a quadratic equation in the form of $a{{x}^{2}}+bx+c=0$ is given as $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . If we assume the roots to be ${{x}_{1}},{{x}_{2}}$ and the factors to be $\left( x-{{x}_{1}} \right)$ and $\left( x-{{x}_{2}} \right)$ .
Now let us find the roots of this equation by using this formula. By doing that we will have $\begin{align}
  & x=\dfrac{-4\pm \sqrt{{{\left( 4 \right)}^{2}}-4\left( 4 \right)}}{2} \\
 & \Rightarrow x=-2 \\
\end{align}$ .
Now we can say that the factored form of this equation is ${{\left( x+2 \right)}^{2}}$ .
Now for verifying this expression we will multiply $x+2$ with itself and see if we are getting the assumed quadratic equation or not.
This is the process for verifying if the quadratic equation is factored perfectly that is by multiplying all the factors.
By doing this we will have
 $\begin{align}
  & \left( x+2 \right)\left( x+2 \right)={{x}^{2}}+2x+2x+4 \\
 & \Rightarrow {{x}^{2}}+4x+4 \\
\end{align}$
Hence we have factorized our expression correctly.
Therefore we can conclude that any quadratic expression factored form can be verified similarly.

Note: While answering questions of this type we should be sure with our concepts that we are going to apply in the process. Similarly we can verify the factorization for any expression for example if we consider a quadratic expression ${{x}^{2}}+10x+25$ if we assume or consider that the factored form is $\left( x+5 \right)\left( x+3 \right)$ then if we verify we will have $\Rightarrow \left( x+5 \right)\left( x+3 \right)={{x}^{2}}+8x+15$ . Hence we can say that the factored form is wrong. The actual factored form is ${{\left( x+5 \right)}^{2}}$ .