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A walkthrough for the entire process of completing the square

**Co-authored by**David Jiaand Carmine Shannon

Last Updated: May 15, 2024Fact Checked

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- Completing the Square |
- Completing the Square (Vertex Form) |
- Solving Quadratic Equations |
- Calculator, Practice Problems, and Answers |
- Video |
- Expert Interview |
- |
- Tips

Completing the square is a helpful technique that allows you to rearrange a quadratic equation into a neat form that makes it easier to visualize or even solve. It’s used to determine the vertex of a parabola and to find the roots of a quadratic equation. If you’re just starting out with completing the square, or if the math isn’t exactly adding up, follow along with these easy steps to become a quadratic whiz.

## Formula for Completing the Square

To complete the square with , subtract C from both sides, then add to both sides. Now the equation can be factored as , where the left side is a perfect square.

Section 1 of 3:

### Completing the Square

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1

**Write down the equation.**Let's say you're working with the following equation:2

**Subtract the last term from both sides of the equation.**To isolate your x terms, move the constant (number without an “x”) to the other side of the equation by subtracting it from both sides.^{[1]}Advertisem*nt

3

**Halve the coefficient of the x term and square it.**Take the number in front of the x (second) term in the equation, divide it by 2, and then square it.^{[2]}In this example, the second term is so and .4

**Add the term to both sides of the equation.**By adding this term to the left side of the equation, you create a polynomial that is a perfect square. To keep the equation balanced, add the number to the right side of the equation, as well.^{[3]}5

**Factor the left side of the equation.**Now that the left side is a perfect square, you can rewrite it as (x + the halved second term)^{2}, or for this example. You can check your math by multiplying it out and seeing if it gives you the first three terms from the last step.^{[4]}6

**Take the square root of both sides.**To solve your equation, undo the square on the left by taking the square root of both sides.^{[5]}.7

**Solve to find the roots of the equation.**When you take the square root of the right side of the equation, the result is both negative and positive, since negative numbers are also positive when they’re squared. To solve this equation, break it into two parts: and .^{[6]}

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Section 2 of 3:

### Writing Vertex Form by Completing the Square

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1

**Write your equation.**For this example, try an equation where the term has a coefficient, like .- Vertex form is a way of writing a quadratic equation with the coordinates of the vertex of the parabola formed by that equation.
- Vertex form is written as where is the vertex of a parabola.
^{[7]}

2

**Move the constant to the right side of the equation.**Isolate the x terms by adding 15 to both sides of the equation.3

**Factor out the coefficient of the squared term from the first 2 terms.**To complete the square, the leading coefficient has to be 1, so factor 3 out of the left side of the equation.^{[8]}.4

**Halve the second term and square it.**The second term, also known as the b term, in the equation, is . First, divide it by two: . Then, square the term: .^{[9]}- Completing the square refers to finding a constant “C” that you can add to the first two terms to make a perfect square trinomial, which can be factored into the expression . The number you get at this step is the constant that completes the square.

5

**Add the term to both sides of the equation.**Use the new term to make a perfect square trinomial by adding it into the parentheses.^{[10]}Since you factored out a 3, multiply the new term by three before adding it to the right side of the equation with the y term.6

**Factor the trinomial as a square and isolate “y.”**Now that you have a perfect square in the parenthesis, factor it out using the halved b term from before, in this case . Write your equation as a factored square, subtract the constant from the y side, and add it to the x side.^{[11]}- If you’re using the vertex form to find the vertex of a parabola (h, k), remember that “h” has to be negative and “k” has to be positive. In this example, the vertex is .
^{[12]}

See AlsoCompleting the Square

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Section 3 of 3:

### Solving Quadratic Equations

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1

**Write down the problem.**Let's say you're working with the following equation: .2

**Move the constant terms to the left side of the equation.**The constant terms are any terms that aren't attached to a variable.^{[13]}In this case, you have 5 on the left side and 6 on the right side, so subtract 6 from both sides of the equation. Now you have3

**Factor out the coefficient of the squared term.**In this case, 3 is the coefficient of the term. Divide each term by 3, then put the terms into parenthesis with a 3 in front. So, , , and . Now the equation is:^{[14]}4

**Divide by the constant you just factored out.**Since you divided each term by 3, it can be removed without impacting the equation. Now you have^{[15]}5

**Move the constant to the right side of the equation.**Isolate the x terms by moving to the other side of the equal sign.6

**Halve the second term and square it.**Next, take the second term, , also known as the*b*term, and halve and square it to complete the square. When you’re done, add it to both sides of the equation to keep it balanced.^{[16]}7

**Write the left side of the equation as a perfect square.**Since you've already used a formula to find the missing term, all you have to do is put x and half of the second coefficient in parentheses and square them, like so: . The equation should now read: .^{[17]}- Note that factoring that perfect square will give you the three terms: .

8

**Take the square root of both sides.**On the left side of the equation, the square root of is just . On the right side, the square root of the denominator, 9, is 3, and the square root of 7 is , so the square root is .^{[18]}- Remember to write because a square root can be positive or negative.

9

**Isolate the variable.**To isolate the variable x, move the constant term over to the right side of the equation. You now have two possible answers for x: . You can leave it at that or find the actual square root of 7 if you need an answer without the radical sign.^{[19]}

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### Calculator, Practice Problems, and Answers

Sample Completing the Square Calculator

Sample Completing the Square Practice Problems

Sample Completing the Square Practice Answers

## Community Q&A

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Question

Why do you halve the b value and then square it? It makes no sense to me.

Community Answer

It does seem strange and arbitrary, but there is a reason for it. The power move is taking the square root of both sides, but you can't simplify the square root of most polynomials. The step you ask about is a setup move to make the power move work. If I have, for example, x^2 + 4x = 5, and take the square root of both sides, nothing happens, it just makes a mess. But if I add 4 to both sides first and take the square root of both sides of x^2 + 4x + 4 = 9, it simplifies to |x+2| = 3 and the quadratic equation is reduced to a linear equation.

**Thanks! We're glad this was helpful.****Thank you for your feedback.**

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Question

What's the completing the square formula if x > 1?

Donagan

Top Answerer

The value of x doesn't matter. The process remains as shown above.

**Thanks! We're glad this was helpful.****Thank you for your feedback.**

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Question

In Part 1 of 2, how did you get 11/9 in Step 8?

Donagan

Top Answerer

Both sides of that equation are being divided by 3 (to get rid of the coefficient of the first term). Dividing the second term (11/3) by 3 gives us 11/9.

**Thanks! We're glad this was helpful.****Thank you for your feedback.**

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## Video

## Tips

Even after you know the quadratic formula, regularly practice completing the square either by proving the quadratic formula or by doing some practice problems. That way you won't forget how to do it when you need it.

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Be sure to put the in front of square roots, otherwise you will only get one answer.

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## Expert Interview

Thanks for reading our article! If you’d like to learn more about math, check out our in-depth interview with David Jia.

## References

- ↑ https://youtu.be/0IFPixtQpYM?t=24
- ↑ https://youtu.be/VvuuRpJbbHE?t=118
- ↑ https://youtu.be/0IFPixtQpYM?t=165
- ↑ https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:more-on-completing-square/a/solving-quadratic-equations-by-completing-the-square
- ↑ https://youtu.be/0IFPixtQpYM?t=311
- ↑ https://www.cuemath.com/algebra/roots-of-quadratic-equation/
- ↑ https://mathbitsnotebook.com/Algebra1/Quadratics/QDVertexForm.html
- ↑ https://mathbitsnotebook.com/Algebra1/Quadratics/QDVertexForm.html
- ↑ http://www.purplemath.com/modules/sqrquad.htm

More References (10)

- ↑ https://mathbitsnotebook.com/Algebra1/Quadratics/QDVertexForm.html
- ↑ https://mathbitsnotebook.com/Algebra1/Quadratics/QDVertexForm.html
- ↑ https://mathbitsnotebook.com/Algebra1/Quadratics/QDVertexForm.html
- ↑ https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:more-on-completing-square/a/solving-quadratic-equations-by-completing-the-square
- ↑ https://www.whatcom.edu/home/showdocument?id=1766
- ↑ https://www.whatcom.edu/home/showdocument?id=1766
- ↑ https://www.whatcom.edu/home/showdocument?id=1766
- ↑ https://www.khanacademy.org/test-prep/sat/x0a8c2e5f:untitled-652/x0a8c2e5f:passport-to-advanced-math-lessons-by-skill/a/gtp--sat-math--article--solving-quadratic-equations--lesson
- ↑ https://www.khanacademy.org/test-prep/sat/x0a8c2e5f:untitled-652/x0a8c2e5f:passport-to-advanced-math-lessons-by-skill/a/gtp--sat-math--article--solving-quadratic-equations--lesson
- ↑ https://www.khanacademy.org/test-prep/sat/x0a8c2e5f:untitled-652/x0a8c2e5f:passport-to-advanced-math-lessons-by-skill/a/gtp--sat-math--article--solving-quadratic-equations--lesson

## About This Article

Co-authored by:

David Jia

Academic Tutor

This article was co-authored by David Jia and by wikiHow staff writer, Carmine Shannon. David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. This article has been viewed 433,310 times.

8 votes - 85%

Co-authors: 32

Updated: May 15, 2024

Views:433,310

Categories: Algebra

Article SummaryX

To complete the square for a standard equation, you'll need to transform the equation to vertex form. Start by factoring out the coefficient of the squared term from the first two terms, then halve the second term and square it. Next, add and subtract this term from the equation. Pull the term you subtracted out of the parentheses, then convert the terms in the parentheses into a perfect square. Lastly, combine the constant terms and write out the equation in vertex form. The vertex form is your answer. If you want to learn more, like how to solve a quadratic function, keep reading the article!

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