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Algebra 1

9.3.3 Solve by Completing the Square

Algebra 19.3.3 Solve by Completing the Square

9.3.3 Solve by Completing the Square

9.3.3 • Solve by Completing the Square

Activity

Here are four equations followed by worked solutions of the equations. Each solution has at least one error.

1. Solve the equation x 2 + 14 x = − 24 x 2 + 14 x = − 24 . Then look at the following worked solution (with errors) and find and describe the error or errors in the worked solution.

x 2 + 14 x = − 24  Step 1  x 2 + 14 x + 28 = 4  Step 2  ( x + 7 ) 2 = 4  Step 3  x + 7 = 2  Step 4  x + 7 = − 2  Step 5  x = − 5  and  x = 9  Step 6  x 2 + 14 x = − 24  Step 1  x 2 + 14 x + 28 = 4  Step 2  ( x + 7 ) 2 = 4  Step 3  x + 7 = 2  Step 4  x + 7 = − 2  Step 5  x = − 5  and  x = 9  Step 6 

Here are four equations followed by worked solutions of the equations. Each solution has at least one error.

a. Where was the error made in the worked solution above and what was the error?

2. Solve the equation x 2 − 10 x + 16 = 0 x 2 − 10 x + 16 = 0 . Then look at the following worked solution (with errors) and find and describe the error or errors in the worked solution.

x 2 − 10 x + 16 = 0  Step 1  x 2 − 10 x + 25 = 9  Step 2  ( x − 5 ) 2 = 9  Step 3  x − 5 = 9  Step 4  x − 5 = − 9  Step 5  x = 14  and  x = − 4  Step 6  x 2 − 10 x + 16 = 0  Step 1  x 2 − 10 x + 25 = 9  Step 2  ( x − 5 ) 2 = 9  Step 3  x − 5 = 9  Step 4  x − 5 = − 9  Step 5  x = 14  and  x = − 4  Step 6 

a. Where was the error made in the worked solution above and what was the error?

3. Solve the equation x 2 + 2.4 x = − 0.8 x 2 + 2.4 x = − 0.8 . Then look at the following worked solution (with errors) and find and describe the error or errors in the worked solution.

x 2 + 2.4 x = − 0.8  Step 1 x 2 + 2.4 x + 1.44 = 0.64 Step 2 ( x + 1.2 ) 2 = 0.64 Step 3 x + 1.2 = 0.8 Step 4 x 2 + 2.4 x = − 0.8  Step 1 x 2 + 2.4 x + 1.44 = 0.64 Step 2 ( x + 1.2 ) 2 = 0.64 Step 3 x + 1.2 = 0.8 Step 4

a. Where was the error made in the worked solution above and what was the error?

4. Solve the equation x 2 − 6 5 x + 1 5 = 0 x 2 − 6 5 x + 1 5 = 0 . Then look at the following worked solution (with errors) and find and describe the error or errors in the worked solution.

x 2 − 6 5 x + 1 5 = 0  Step 1  x 2 − 6 5 x + 9 25 = 9 25  Step 2  ( x − 3 5 ) 2 = 9 25  Step 3  x − 3 5 = 3 5  Step 4  x − 3 5 = − 3 5  Step 5  x = 6 5  and  x = 0  Step 6  x 2 − 6 5 x + 1 5 = 0  Step 1  x 2 − 6 5 x + 9 25 = 9 25  Step 2  ( x − 3 5 ) 2 = 9 25  Step 3  x − 3 5 = 3 5  Step 4  x − 3 5 = − 3 5  Step 5  x = 6 5  and  x = 0  Step 6 

a. Where was the error made in the worked solution above and what was the error?

Self Check

Examine the work shown. Which of the following is the error that took place when the equation was solved by completing the square?

x 2 − 6 x − 7 = 0

x 2 − 6 x = 7

x 2 − 6 x + 9 = 16

( x − 3 ) 2 = 16

x − 3 = 4

x = 7

  1. This equation cannot be solved by completing the square.
  2. When completing the square, the wrong number was found for ( b 2 ) 2 .
  3. When taking the square root, only the positive solution was included.
  4. The 7 should have been subtracted from each side.

Additional Resources

Spotting the Error When Solving Quadratics

To find the error when solving quadratics, it is helpful to first solve the problem on your own.

Solve the quadratic equation by completing the square and finding its zeros. Identify the error in the worked solution below.

x 2 + 10 x − 34 = 0  Step 1  x 2 + 10 x = 34  Step 2  x 2 + 10 x + 25 = 34 − 25  Step 3  ( x + 5 ) 2 = 9  Step 4  x + 5 = ± 3  Step 5  x = − 8 , x = − 2  Step 6  x 2 + 10 x − 34 = 0  Step 1  x 2 + 10 x = 34  Step 2  x 2 + 10 x + 25 = 34 − 25  Step 3  ( x + 5 ) 2 = 9  Step 4  x + 5 = ± 3  Step 5  x = − 8 , x = − 2  Step 6 

Now, find the error by working through the problem and comparing steps:

x 2 + 10 x − 34 = 0  Step 1  x 2 + 10 x = 34  Step 2  x 2 + 10 x + 25 = 34 + 25  Step 3  x 2 + 10 x − 34 = 0  Step 1  x 2 + 10 x = 34  Step 2  x 2 + 10 x + 25 = 34 + 25  Step 3 

In Step 3, 25 was added to the left side of the equation, but 25 was SUBTRACTED from the right side of the equation. Equations must remain balanced, and 25 should be added to each side of the equation.

Try it

Spotting the Error When Solving Quadratics

Examine the steps below and identify the error in solving the quadratic equation by completing the square:

x 2 + 8 x + 7 = 0  Step 1  x 2 + 8 x = − 7  Step 2  x 2 + 8 x + 16 = − 7 + 16  Step 3  ( x + 4 ) 2 = 9  Step 4  x + 4 = 3  Step 5  x = − 1  Step 6  x 2 + 8 x + 7 = 0  Step 1  x 2 + 8 x = − 7  Step 2  x 2 + 8 x + 16 = − 7 + 16  Step 3  ( x + 4 ) 2 = 9  Step 4  x + 4 = 3  Step 5  x = − 1  Step 6 

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