table of contents solving quadratic equations – completing the square it is assumed you have...

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Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete the square on a binomial. Completing the Square is a method that can be used to solve any quadratic equation The previous methods of factoring and square roots to solve quadratics were limited, in that there are many quadratic equations that neither method would solve.

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Page 1: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

Solving Quadratic Equations – Completing the Square

• It is assumed you have already watched the slideshow demonstrating how to complete the square on a binomial.

• Completing the Square is a method that can be used to solve any quadratic equation

• The previous methods of factoring and square roots to solve quadratics were limited, in that there are many quadratic equations that neither method would solve.

Page 2: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

Solving a Quadratic Equation by Completing the Square

2x bx k 1. Write the equation in the form

2. Complete the square

3. Factor the perfect square trinomial

4. Solve using the square root method

Page 3: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

• Example 1:

Note that the equation is already in the proper form.

Complete the square.

Solve the equation. 2 6 7x x

2 6 9 7 9x x

2x bx k

2

1

26 3

3 9

Add this value to both sides of the equation.

Page 4: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

Complete the problem using the square root method.

Factor 23 16x

3 16

3 4

3 4

1,7

x

x

x

x

2 6 9 7 9x x

Page 5: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

• Example 2:

Write the equation in the proper form.

Complete the square. Be sure to add the same value to both sides of the equation.

Solve the equation. 2 4 8x x

2 8 16 4 16x x

2 8 4x x

Page 6: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

Complete the problem using the square root method.

Factor

4 20

4 2 5

4 2 5

x

x

x

24 20x

2 8 16 4 16x x

Page 7: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

• Example 3:

Write the equation in the proper form.

Complete the square and add to both sides.

Solve the equation. 210 7x x

2 49 497 10

4 4x x

2 7 10x x

Simplify 2 49 89

74 4

x x

Page 8: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

Complete the problem using the square root method.

Factor

27 89

2 4x

7 89

2 4x

2 49 897

4 4x x

Page 9: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

7 89

2 2x

7 89

2x

Use the common denominator to combine the fractions.

7 89

2 2x

Page 10: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

• Example 4:

Write the equation in the proper form.

Divide by 3 which is the coefficient of the quadratic term.

Solve the equation. 23 6 45 0x x

23 6 45x x

2 2 15x x

To be able to complete the square, the coefficient of the quadratic term must be 1.

Page 11: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

Complete the square. 2 2 1 15 1x x

Factor 21 16x

Complete the problem using the square root method.

1 16

1 4

3,5

x

x

x

2 2 15x x

Page 12: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

• Example 5:

Write the equation in the proper form.

Complete the square.

Solve the equation. 2 12 40 0x x

2 12 40x x

2 12 36 40 36x x

Factor 26 4x

Page 13: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents

Complete the problem using the square root method.

26 4

6 4

6 2

6 2

x

x

x i

x i

Page 14: Table of Contents Solving Quadratic Equations – Completing the Square It is assumed you have already watched the slideshow demonstrating how to complete

Table of Contents