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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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
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• 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.
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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
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• 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
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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
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• 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
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Complete the problem using the square root method.
Factor
27 89
2 4x
7 89
2 4x
2 49 897
4 4x x
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7 89
2 2x
7 89
2x
Use the common denominator to combine the fractions.
7 89
2 2x
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• 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.
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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
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• 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
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Complete the problem using the square root method.
26 4
6 4
6 2
6 2
x
x
x i
x i
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