### About Completing the Square:

We previously learned how to solve quadratic equations by factoring. In many cases, we must utilize a different method. When this occurs, we can turn to a method known as completing the square. This method creates a perfect square trinomial on one side and sets it equal to a constant on the other. We can then solve using the square root property.

Test Objectives

- Demonstrate the ability to use the square root property
- Demonstrate the ability to solve a quadratic equation by completing the square
- Demonstrate the ability to solve a quadratic equation with a complex solution

#1:

Instructions: Solve each using the square root property.

a) $$5r^2 - 8 = 72$$

b) $$3x^2 - 4 = 41$$

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#2:

Instructions: Solve each using the square root property.

a) $$(x - 4)^2 - 7 = -6$$

b) $$(x + 7)^2 + 2 = 12$$

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#3:

Instructions: Solve each by completing the square.

a) $$13x^2 - x - 35 = 7x^2$$

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#4:

Instructions: Solve each by completing the square.

a) $$12p^2 - 76 = -14p$$

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#5:

Instructions: Solve each by completing the square.

a) $$-b^2 - 5b +5 = -3b^2$$

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Written Solutions:

#1:

Solutions:

a) $$r = 4 \hspace{.5em} or \hspace{.5em} r = -4$$

b) $$x = \sqrt{15} \hspace{.5em} or \hspace{.5em} x = -\sqrt{15}$$

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#2:

Solutions:

a) $$x = 5 \hspace{.5em} or \hspace{.5em} x = 3$$

b) $$x = -7 + \sqrt{10} \hspace{.5em} or \hspace{.5em} x = -7 - \sqrt{10}$$

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#3:

Solutions:

a) $$x = \frac{5}{2} \hspace{.5em} or \hspace{.5em} x = -\frac{7}{3}$$

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#4:

Solutions:

a) $$p = 2 \hspace{.5em} or \hspace{.5em} p = -\frac{19}{6}$$

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#5:

Solutions:

a) $$b = \frac{5 + i\sqrt{15}}{4} \hspace{.5em} or \hspace{.5em} b = \frac{5 - i\sqrt{15}}{4}$$