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Algebra 1

9.3.2 Using Completing the Square to Solve Equations

Algebra 19.3.2 Using Completing the Square to Solve Equations

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Activity

Completing the square can be used to solve quadratic equations.

Example

Solve x2+8x2=0x2+8x2=0.

To complete the square to solve an equation, follow these steps:

Step 1 - Write the equation in the form ax2+bx=cax2+bx=c.

x2+8x=2x2+8x=2

Step 2 - Complete the square by finding (b2)2(b2)2.

(b2)2=(82)2=16(b2)2=(82)2=16

Step 3 - Add the value to both sides of the equation.

x2+8x+16=2+16x2+8x+16=2+16

Step 4 - Factor the perfect square trinomial and simplify the constants.

(x+4)2=14(x+4)2=14

Step 5 - Take the square root of each side (remember the plus-minus sign with the constant).

(x+4)2=14(x+4)2=14

x+4=±14x+4=±14

Step 6 - Write two separate equations and solve each.

x+4=14x+4=14  or  x+4=14x+4=14 x=4+14x=4+14  or  x=414x=414

Solve each equation by completing the square. In part a, select the value of (b2)2(b2)2 that will complete the square. In part b, choose the two roots.

1.

Enter the value:

( x 3 ) ( x + 1 ) = 5 ( x 3 ) ( x + 1 ) = 5

What is the correct value of (b2)2(b2)2 that will complete the square?

2.

Select two roots that solve the quadratic equation.

3.

x 2 + 1 2 x = 3 16 x 2 + 1 2 x = 3 16

What is the correct value of (b2)2(b2)2 that will complete the square?

4.

Select two roots that solve the quadratic equation.

5.

x 2 + 3 x + 8 4 = 0 x 2 + 3 x + 8 4 = 0

What is the correct value of (b2)2(b2)2 that will complete the square?

6.

Select two roots that solve the quadratic equation.

7.

( 7 x ) ( 3 x ) + 3 = 0 ( 7 x ) ( 3 x ) + 3 = 0

What is the correct value of (b2)2(b2)2 that will complete the square?

8.

Select two roots that solve the quadratic equation.

9.

x 2 + 1.6 x + 0.63 = 0 x 2 + 1.6 x + 0.63 = 0

What is the correct value of (b2)2(b2)2 that will complete the square?

10.

Select two roots that solve the quadratic equation.

Video: Solve by Completing the Square

Watch the following video to learn more about solving to complete the square.

Self Check

After solving using completing the square, what are the zeros of the quadratic equation x 2 + 5 x + 9 4 = 0 ?
  1. x = 1 2
  2. x = 1 2 , x = 9 2
  3. x = 25 4
  4. x = 1 4 , x = 19 4

Additional Resources

Completing the Square with Fractions

Completing the square can be a useful method for solving quadratic equations in cases in which it is not easy to rewrite an expression in factored form. For example, let's solve this equation:

x2+5x754=0x2+5x754=0

First, we'll add 754754 to each side to make things easier on ourselves.

x2+5x754+754=0+754x2+5x754+754=0+754

x2+5x=754x2+5x=754

To complete the square, take 1212 of the coefficient of the linear term 5, which is 5252, and square it, which is 254254. Add this to each side:

x2+5x+254=754+254x2+5x+254=754+254

x2+5x+254=1004x2+5x+254=1004

Notice that 10041004 is equal to 25 and rewrite it:

x2+5x+254=25x2+5x+254=25

Since the left side is now a perfect square, let's rewrite it:

(x+52)2=25(x+52)2=25

For this equation to be true, one of these equations must be true:

x+52=5x+52=5 or x+52=5x+52=5

To finish up, we can subtract 5252 from each side of the equal sign in each equation.

x=552x=552 or x=552x=552

x=52x=52 or x=152x=152

x=212x=212 or x=712x=712

It takes some practice to become proficient at completing the square, but it makes it possible to solve many more equations than you could by methods you learned previously.

Try it

Try It: Completing the Square with Fractions

Solve x2+3x4=0x2+3x4=0 using completing the square, then find the zeros of the quadratic equation.

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