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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 | |
|---|---|---|
|
|
One way to solve the quadratic equation
Then, knowing that
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.
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
When we expand a squared factor like
Likewise, if we expand
To complete the square is essentially to make one side of the equation have the same structure as
The square roots of the expression on the right are the values of
Once
The solution is the quadratic formula!