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Introduction
Completing the Square helps you connect a mathematical rule to a result you can explain and check. Add the square of half the linear coefficient to both sides. Rewrite as a perfect square and solve both square-root cases. This lesson belongs to quadratic expressions and equations and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in completing the square.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Use signed arithmetic, factoring, square roots, and function notation.
Previous lesson: Square Root Method
Key mathematical vocabulary
- coefficient
- A factor multiplying a variable term.
- equation
- A statement that two expressions are equal.
- function
- A rule assigning one output to each permitted input.
- domain
- The set of permitted inputs.
- range
- The set of outputs produced.
Formulas and rules
- f(x)=ax²+bx+c, a≠0
- f(x)=a(x−h)²+k
- x=(−b±√(b²−4ac))/(2a)
Quadratic functions and equations
Add the square of half the linear coefficient to both sides. Rewrite as a perfect square and solve both square-root cases.
A quadratic function has the form f(x)=ax²+bx+c with a≠0. Its graph is a parabola. Vertex form f(x)=a(x−h)²+k identifies the vertex (h,k), axis x=h, and opening direction from the sign of a.
Factored form connects roots with linear factors: a(x−r₁)(x−r₂) has zeros r₁ and r₂. Solve real quadratic equations by factoring, square roots, completing the square, or the quadratic formula. The discriminant b²−4ac determines whether there are two, one repeated, or no real roots. A quadratic’s mathematical domain is all real numbers, while its range is bounded by its extreme value.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. Move the constant: x²−18x=63.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Substituting either -3 or 21 into the original polynomial gives 0; both roots are valid.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Solve x² − 18x + (-63) = 0 over the real numbers.
- Move the constant: x²−18x=63.
- Half the x-coefficient is −9; its square is 81. Add 81 to BOTH sides: (x−9)²=144.
- Take both square roots: x−9=±12, so x=-3 or x=21.
Answer: -3 or 21
Verify: Substituting either -3 or 21 into the original polynomial gives 0; both roots are valid.
Avoid this mistake: Dropping the ± when solving a squared equation.
Example 2 · Basic
Solve x² − 20x + (-44) = 0 over the real numbers.
- Move the constant: x²−20x=44.
- Half the x-coefficient is −10; its square is 100. Add 100 to BOTH sides: (x−10)²=144.
- Take both square roots: x−10=±12, so x=-2 or x=22.
Answer: -2 or 22
Verify: Substituting either -2 or 22 into the original polynomial gives 0; both roots are valid.
Avoid this mistake: Reading the vertex sign incorrectly from (x−h)².
Example 3 · Intermediate
Solve x² − 22x + (-23) = 0 over the real numbers.
- Move the constant: x²−22x=23.
- Half the x-coefficient is −11; its square is 121. Add 121 to BOTH sides: (x−11)²=144.
- Take both square roots: x−11=±12, so x=-1 or x=23.
Answer: -1 or 23
Verify: Substituting either -1 or 23 into the original polynomial gives 0; both roots are valid.
Avoid this mistake: Assuming every parabola crosses the x-axis twice.
Example 4 · Advanced
Solve x² − 4x + (-165) = 0 over the real numbers.
- Move the constant: x²−4x=165.
- Half the x-coefficient is −2; its square is 4. Add 4 to BOTH sides: (x−2)²=169.
- Take both square roots: x−2=±13, so x=-11 or x=15.
Answer: -11 or 15
Verify: Substituting either -11 or 15 into the original polynomial gives 0; both roots are valid.
Avoid this mistake: Dropping the ± when solving a squared equation.
Example 5 · Challenge
Solve x² − 6x + (-160) = 0 over the real numbers.
- Move the constant: x²−6x=160.
- Half the x-coefficient is −3; its square is 9. Add 9 to BOTH sides: (x−3)²=169.
- Take both square roots: x−3=±13, so x=-10 or x=16.
Answer: -10 or 16
Verify: Substituting either -10 or 16 into the original polynomial gives 0; both roots are valid.
Avoid this mistake: Reading the vertex sign incorrectly from (x−h)².
Common mistakes and corrections
- Dropping the ± when solving a squared equation.
- Reading the vertex sign incorrectly from (x−h)².
- Assuming every parabola crosses the x-axis twice.
Real-world applications
- Model the height of a thrown object over its physical time interval. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Find maximum area or minimum cost from a quadratic model. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Add the square of half the linear coefficient to both sides. Rewrite as a perfect square and solve both square-root cases.
A quadratic function has the form f(x)=ax²+bx+c with a≠0. Its graph is a parabola. Vertex form f(x)=a(x−h)²+k identifies the vertex (h,k), axis x=h, and opening direction from the sign of a.
Frequently asked questions
What should I know before completing the square?
Use signed arithmetic, factoring, square roots, and function notation.
How can I check my work?
Substituting either -3 or 21 into the original polynomial gives 0; both roots are valid. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Dropping the ± when solving a squared equation. Reading the vertex sign incorrectly from (x−h)². Assuming every parabola crosses the x-axis twice.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Model the height of a thrown object over its physical time interval.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Add the square of half the linear coefficient to both sides. Rewrite as a perfect square and solve both square-root cases.
- Dropping the ± when solving a squared equation.
- Reading the vertex sign incorrectly from (x−h)².
- Assuming every parabola crosses the x-axis twice.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
Continue learning
Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials