--- title: "Completing the Square" description: "Write a quadratic as (x + p)² + q by halving the x coefficient and correcting the constant, then solve it by taking the square root of both sides." canonical: https://lightmysky.com/learn/mathematics/completing-the-square-mt_rHmCfzB4Fm source: https://lightmysky.com/learn/mathematics/completing-the-square-mt_rHmCfzB4Fm.md retrieved: 2026-09-03 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Completing the Square Write a quadratic as (x + p)² + q by halving the x coefficient and correcting the constant, then solve it by taking the square root of both sides. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/completing-the-square-mt_rHmCfzB4Fm ## Ready when they can - Write x² + 6x + 1 as (x + 3)² - 8 - Solve from the completed square by taking the square root and keeping both signs - Complete the square when the x² coefficient is not 1 by taking it out first ## Lesson: Completing the Square You have seen perfect square brackets before: (x + 5) squared expands to x squared + 10x + 25, and you can spot the pattern and take it apart. This stop goes the other way. It forces any quadratic into the shape (x + p) squared + q, even when the constant does not fit yet. That finished form is the key, because (x + p) squared + q = 0 can be solved by taking a square root, and the finished form (x + p) squared + q reads off the lowest point of the curve. *(drawing: The moves of completing the square, in the order they happen.)* Here is why the moves work. In x squared + 6x, the x coefficient is 6. Half of 6 is 3, and 3 squared is 9. The perfect square (x + 3) squared expands to x squared + 6x + 9, so it matches the first two terms exactly if we add 9. But adding 9 changes the value of the expression, so we also subtract 9 to keep it the same. Add and subtract the same number and nothing has really changed, but now the first three terms form a perfect square. **Example.** Write x squared + 6x + 1 as (x + p) squared + q. Halve the x coefficient: half of 6 is 3. Square it: 9. Add and subtract 9 in the middle: x squared + 6x + 9 - 9 + 1. The first three terms are (x + 3) squared, so the expression becomes (x + 3) squared - 8. Check by expanding: (x + 3) squared is x squared + 6x + 9, and 9 - 8 is 1, so we are back to x squared + 6x + 1. **Example.** Now use the finished form to solve. Solve x squared + 6x + 1 = 0. We just wrote the left side as (x + 3) squared - 8, so the equation is (x + 3) squared - 8 = 0, which rearranges to (x + 3) squared = 8. Take the square root of both sides and keep both signs: x + 3 = the square root of 8, or x + 3 = minus the square root of 8. So x = -3 plus or minus the square root of 8. **Example.** Now the x squared term does not start at 1. Write 2x squared + 8x + 3 in the form a(x + p) squared + q. Take out the factor 2 from the first two terms: 2(x squared + 4x) + 3. Inside the bracket, halve the x coefficient: half of 4 is 2, and 2 squared is 4. Add and subtract 4 inside: 2(x squared + 4x + 4 - 4) + 3. The bracket is (x + 2) squared, so this is 2((x + 2) squared - 4) + 3, which expands to 2(x + 2) squared - 8 + 3, that is 2(x + 2) squared - 5. The finished form reads off the lowest point of the curve as well. In (x + p) squared + q, the square part is never negative, so the smallest value the expression can take is q, and it happens where the square is zero, that is at x = -p. For (x + 3) squared - 8 the minimum is -8 at x = -3, and for 2(x + 2) squared - 5 it is -5 at x = -2. **Recap.** Halve the x coefficient, square it, add and subtract that number to build a perfect square, then solve by taking the square root with both signs. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Rearranging Formulae When the Subject Appears Twice](https://lightmysky.com/learn/mathematics/rearranging-formulae-when-the-subject-appears-twice-mt_Velmpnxfcj) - [Difference of Two Squares and Perfect Square Trinomials](https://lightmysky.com/learn/mathematics/difference-of-two-squares-and-perfect-square-trinomials-mt_GSpsHuserT) ## Opens up - [The Quadratic Formula](https://lightmysky.com/learn/mathematics/the-quadratic-formula-mt_C-4brnRYHY) - [The Equation of a Circle](https://lightmysky.com/learn/mathematics/the-equation-of-a-circle-mt_YARqmi_ioe)