Activity
One technique for solving quadratic equations is called completing the square. Here are two examples of how Diego and Mai completed the square to solve the same equation.
Diego | Mai |
Study the worked examples. Then, try solving these equations by completing the square for questions 1 – 5.
Compare your answer:
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Are you ready for more?
Extending Your Thinking
Here is a diagram made of a square and two congruent rectangles. Its total area is square units.
Diagram A
What is the length of the unlabeled side of each of the two rectangles in Diagram A?
Enter the length of the unlabeled side of each of the two rectangles.
Compare your answer:
17.5
In Diagram B, we add lines to make the figure a square. The area of the square becomes a quadratic equation.
Calculate the equation, then put that equation in the standard form to answer questions a and b.
Using the standard form to represent the area of the square in Diagram B, what is the value of the coefficient ?
Enter the value of the coefficient .
Compare your answer:
35
Using the standard form to represent the area of the square in Diagram B, what is the value of the constant ?
Enter the value of the constant .
Compare your answer:
306.25
How is the process of finding the area of the entire figure like the process of building perfect squares for expressions like ?
Enter your explanation of how the two processes are alike.
Compare your answer:
First, we had to figure out what number was half of 35 (which is 17.5) to find the side length of the rectangles. Then, we had to square the number 17.5 to find the area of the missing piece.
Video: Solve Equations by Completing the Square
Watch the following video to learn more about solving equations by completing the square.
Self Check
Additional Resources
Solving Quadratic Equations by Completing the Square
Example
Solve by completing the square.
Step 1 - Isolate the variable terms on one side and place the constant term on the other. This equation has all the variables on the left.
Step 2 - Find (, the number to complete the square. Add to both sides of the equation. Take half of 8 and square it.
Add 16 to BOTH sides of the equation.
Step 3 - Factor the perfect square trinomial as a binomial square.
Substitute the binomial square term for the trinomial.
Step 4 - Use the square root property.
Step 5 - Simplify the radical and then solve the two resulting equations.
Step 6 - Check the solutions. Put each answer in the original equation to check.
Substitute .
Substitute .
Try it
Try It: Solving Quadratic Equations by Completing the Square
Solve by completing the square.
Here is how to solve this quadratic using completing the square:
Step 1 - Isolate the constants on one side of the equation.
Step 2 - Find . Add this number to both sides of the equation.
,
Step 3 - Factor the perfect square trinomial.
Step 4 - Use the square root property.
Step 5 - Separate into two equations and solve.
Step 6 - Check into the original equation.