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
Completing the Square Practice Test:

#1:

Instructions: solve each equation.

$$a)\hspace{.2em}x^2 - 4x - 32=0$$

$$b)\hspace{.2em}x^2 - 4x - 60=0$$


#2:

Instructions: solve each equation.

$$a)\hspace{.2em}x^2 - 10x - 36=0$$

$$b)\hspace{.2em}3x^2 + 6x - 70=-10$$


#3:

Instructions: solve each equation.

$$a)\hspace{.2em}x^2 - 4x + 48=-5$$

$$b)\hspace{.2em}2x^2 + 4x + 4=10$$


#4:

Instructions: solve each equation.

$$a)\hspace{.2em}4x^2 + 73=6 - 2x$$

$$b)\hspace{.2em}8x^2 - 10x + 5=-8x$$


#5:

Instructions: solve each equation.

$$a)\hspace{.2em}69 - 17x=4x - 2x^2$$

$$b)\hspace{.2em}10x^2 - 4x=142$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}x=-4, 8$$

$$b)\hspace{.2em}x=-6, 10$$


#2:

Solutions:

$$a)\hspace{.2em}x=5 \pm \sqrt{61}$$

$$b)\hspace{.2em}x=-1 \pm \sqrt{21}$$


#3:

Solutions:

$$a)\hspace{.2em}x=2 \pm 7i$$

$$b)\hspace{.2em}x=-3, 1$$


#4:

Solutions:

$$a)\hspace{.2em}x=\frac{-1 \pm i\sqrt{267}}{4}$$

$$b)\hspace{.2em}x=\frac{1 + i\sqrt{39}}{8}$$


#5:

Solutions:

$$a)\hspace{.2em}x=\frac{21 \pm i\sqrt{111}}{4}$$

$$b)\hspace{.2em}x=\frac{1 \pm 2\sqrt{89}}{5}$$