Here are some perfect squares in factored and standard forms, and expressions showing how the two forms are related.
Complete the table.
factored form
standard form
Look at the expression in the last row of the table. If is equivalent to , how are , and related to and ?
19.2
Activity
Complete the Square Using a Placeholder
One way to solve the quadratic equation is by completing the square. A partially solved equation is shown here. Study the steps.
Then, knowing that is a placeholder for , continue to solve for without evaluating any part of the expression. Be prepared to explain each step.
Explain how the solution is related to the quadratic formula.
19.3
Activity
Decoding the Quadratic Formula
Here is one way to make sense of how the quadratic formula came about. Study the derivation until you can explain what happens in each step. Record your explanation next to each step.
Student Lesson Summary
Recall that any quadratic equation can be solved by completing the square. The quadratic formula is essentially what we get when we put all the steps taken to complete the square for into a single expression.
When we expand a squared factor like , the result is . Notice how appears in two places. If we replace with another letter, like , we have , which is a recognizable perfect square.
Likewise, if we expand , we have . Replacing with gives , also a recognizable perfect square.
To complete the square is essentially to make one side of the equation have the same structure as . Substituting a letter for makes it easier to see what is needed to complete the square. Let’s complete the square for !
Start by subtracting from each side.
Next, let’s multiply both sides by , which is allowed because . On the left, this gives , a perfect square for the coefficient of .
can be written , and can be written .
Let’s replace with the letter .
is the constant term that completes the square, so let’s add to each side.
The left side is now a perfect square and can be written as a squared factor.
The square roots of the expression on the right are the values of .
Once is isolated, we can write in its place and solve for .
The solution is the quadratic formula!
Glossary
None
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