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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
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.
- 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
$$a)\hspace{.2em}x^2 - 10x - 36=0$$
$$b)\hspace{.2em}3x^2 + 6x - 70=-10$$
$$a)\hspace{.2em}x^2 - 4x + 48=-5$$
$$b)\hspace{.2em}2x^2 + 4x + 4=10$$
$$a)\hspace{.2em}4x^2 + 73=6 - 2x$$
$$b)\hspace{.2em}8x^2 - 10x + 5=-8x$$
$$a)\hspace{.2em}69 - 17x=4x - 2x^2$$
$$b)\hspace{.2em}10x^2 - 4x=142$$
$$a)\hspace{.2em}x=-4, 8$$
$$b)\hspace{.2em}x=-6, 10$$
Watch the Step by Step Video Solution
$$a)\hspace{.2em}x=5 \pm \sqrt{61}$$
$$b)\hspace{.2em}x=-1 \pm \sqrt{21}$$
$$a)\hspace{.2em}x=2 \pm 7i$$
$$b)\hspace{.2em}x=-3, 1$$
$$a)\hspace{.2em}x=\frac{-1 \pm i\sqrt{267}}{4}$$
$$b)\hspace{.2em}x=\frac{1 \pm i\sqrt{39}}{8}$$
$$a)\hspace{.2em}x=\frac{21 \pm i\sqrt{111}}{4}$$
$$b)\hspace{.2em}x=\frac{1 \pm 2\sqrt{89}}{5}$$
More Examples of Completing the Squares
In my opinion, the “most important” usage of completing the square method is when we solve quadratic equations. In fact, the Quadratic Formula that we utilize to solve quadratic equations is derived using the technique of completing the square. Here is my lesson on Deriving the Quadratic Formula .
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 side of the equation, while keeping the x -terms on the left. I can do that by subtracting both sides by 14 .
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.
Take the square roots of both sides of the equation to eliminate the power of 2 of the parenthesis. Make sure that you attach the plus or minus symbol to the constant term (right side of the equation).
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.
Therefore, the final answers are {x_1} = 7 and {x_2} = 2 . You may back-substitute these two values of x from the original equation to check.
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.
Divide 8 by 2 and square it.
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.
Divide the entire equation by the coefficient of the {x^2} term which is 6 . Reduce the fraction to its lowest term.
Identify the coefficient of the linear term.
Divide this coefficient by 2 and square it.
![[(23/2)/2]^2 = 529/16](https://www.chilimath.com/wp-content/uploads/2019/01/compsqrme-ex3e.png "practice problems for solving quadratic equations by completing the square")
Add this output to both sides of the equation. Be careful when adding or subtracting fractions.
Express the trinomial on the left side as a perfect square binomial. Then solve the equation by first taking the square roots of both sides. Don’t forget to attach the plus or minus symbol to the square root of the constant term on the right side.
Finish this off by subtracting both sides by {{{23} \over 4}} . You should obtain two values of “ x ” because of the “plus or minus”.
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.
![[(2/3)/2]^2=1/9](https://www.chilimath.com/wp-content/uploads/2019/01/compsqrme-ex4c.png "practice problems for solving quadratic equations by completing the square")
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.
Step 5: Take the square roots of both sides of the equation. Make sure that you attach the “plus or minus” symbol to the square root of the constant on the right side. Simplify the radical.
Step 6: Solve for x by subtracting both sides by {1 \over 3} . You should have two answers because of the “plus or minus” case.
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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.