If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

## Unit 14: Lesson 8

- Solve by completing the square: Integer solutions
- Solve by completing the square: Non-integer solutions
- Worked example: completing the square (leading coefficient ≠ 1)
- Solving quadratics by completing the square: no solution
- Proof of the quadratic formula
- Solving quadratics by completing the square
- Completing the square review
- Quadratic formula proof review

## Solve equations by completing the square

- (Choice A) x = 2 ± 5 x=2 \pm 5 x = 2 ± 5 x, equals, 2, plus minus, 5 A x = 2 ± 5 x=2 \pm 5 x = 2 ± 5 x, equals, 2, plus minus, 5
- (Choice B) x = − 2 ± 5 x=-2 \pm 5 x = − 2 ± 5 x, equals, minus, 2, plus minus, 5 B x = − 2 ± 5 x=-2 \pm 5 x = − 2 ± 5 x, equals, minus, 2, plus minus, 5
- (Choice C) x = 2 ± 5 x=2 \pm \sqrt{5} x = 2 ± 5 x, equals, 2, plus minus, square root of, 5, end square root C x = 2 ± 5 x=2 \pm \sqrt{5} x = 2 ± 5 x, equals, 2, plus minus, square root of, 5, end square root
- (Choice D) x = − 2 ± 5 x=-2 \pm \sqrt{5} x = − 2 ± 5 x, equals, minus, 2, plus minus, square root of, 5, end square root D x = − 2 ± 5 x=-2 \pm \sqrt{5} x = − 2 ± 5 x, equals, minus, 2, plus minus, square root of, 5, end square root

## Completing the Square Practice Test

- 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

Instructions: solve each equation.

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

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

Watch the Step by Step Video Solution | View the Written Solution

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

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

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

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

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

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

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

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

Watch the Step by Step Video Solution

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

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

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

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

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

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

## More Examples of Completing the Squares

## Applications of Completing the Square Method

Example 1 : Solve the equation below using the method of completing the square.

Next, identify the coefficient of the linear term (just the x -term) which is

Take that number, divide by 2 and square it.

Add {{81} \over 4} to both sides of the equation, and then simplify.

Express the trinomial on the left side as a square of binomial.

Solve for “ x ” by adding both sides by {9 \over 2} .

Find the two values of “ x ” by considering the two cases: positive and negative.

Example 2 : Solve the equation below using the method of completing the square.

Subtract 2 from both sides of the quadratic equation to eliminate the constant on the left side.

Add 16 to both sides of the equation.

Express the left side as square of a binomial.

Take square roots of both sides.

Example 3 : Solve the equation below using the technique of completing the square.

Eliminate the constant - 36 on the left side by adding 36 to both sides of the quadratic equation.

Identify the coefficient of the linear term.

Divide this coefficient by 2 and square it.

Add this output to both sides of the equation. Be careful when adding or subtracting fractions.

The final answers are {x_1} = {1 \over 2} and {x_2} = - 12 .

Example 4 : Solve the equation below using the technique of completing the square.

Step 1: Eliminate the constant on the left side, and then divide the entire equation by - \,3 .

Step 2: Take the coefficient of the linear term which is {2 \over 3} . Divide it by 2 and square it.

Step 3: Add the value found in step #2 to both sides of the equation. Then combine the fractions.

Step 4: Express the trinomial on the left side as square of a binomial.

## IMAGES

## VIDEO

## COMMENTS

Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more.

Problem · (Choice A). x = 2 ± 5 x=2 \pm 5 x=2±5x, equals, 2, plus minus, 5 · (Choice B). x = − 2 ± 5 x=-2 \pm 5 x=−2±5x, equals, minus, 2, plus minus, 5 · (

About Completing the Square: We previously learned how to solve quadratic equations by factoring. In many cases, we must utilize a different method.

“Completing the square” is another method of solving quadratic equations. It allows trinomials to be factored into two identical factors. Example: 2 +

Using the square root property it is possible to solve any quadratic equation written in the form. ( x + b ). 2. = c . The key to setting these problems

Write a solution to the following problems. ... What value needs to be placed in the box to complete the square? ... Solve by completing the square.

Solving Quadratic Equations By Completing the Square. Solve each equation by completing the square. 1) p. 2 + 14p − 38 = 0. 2) v. 2 + 6v − 59 = 0. 3) a.

Practice questions · x 2 + 8 x − 9 = 0 x^2 + 8x - 9 = 0 x2+8x−9=0 · 3 x 2 + 12 x = 0 3x^2 + 12x = 0 3x2+12x=0 · x 2 + 4 x = − 4 x^2 + 4x = -4 x2

Applications of Completing the Square Method ... Example 1: Solve the equation below using the method of completing the square. ... Move the constant to the right

This video continues with solving equations by completing the square. More example problems are covered and discussed.