Compare the steps needed to complete each task. How do both processes relate? Is one contained in the other?
See solution.
Practice makes perfect
We will start by reviewing both methods of solving a quadratic equation to understand how they relate and how they differ.
Solving By Using Square Roots
Solving by using square roots can be used when the quadratic equation is of the form shown below.
x^2 = k
In this case we proceed by calculating the square root of both sides of the equation and, depending on the value of k, we can have different solutions.
If k>0, we have two real-numbers solutions. These are x=sqrt(k) and x=- sqrt(k).
If k=0, we have one real-number solution. This is x=0.
If k<0, we have no real-number solutions.
Solving By Completing the Square
This method can be used to solve any quadratic equation. The procedure is as follows.
Calculate the square roots of both sides and isolate x.
Following the first three steps lets us rewrite the original equation as shown below.
x^2+bx = c
⇕
( x + ( b2)^2 )^2 = c + ( b2)^2
If we think about x + ( b2)^2 as a new variable, x', we can see that this equation is exactly of the form needed to be solved by using square roots.
( x + ( b2)^2 )^2 &= c + ( b2)^2 [2em]
(x')^2 &= k
Conclusions
We can compare both methods and summarize the information presented above as shown below.
Solving by using square roots is only possible for equations of the form x^2= k, while completing the square can be used to solve any quadratic equation.
The process of completing the square is a way to rewrite the quadratic equation so that it can be solved by using square roots. Therefore, solving by using square roots is a smaller task contained in the process of solving by completing the square.
