Completing the Square

Learn and apply completing the square using equations, examples, and interpretation.

Lesson 4 of 53 parts Practice Flashcards

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Creating a Perfect Square

Objective: Explain and use creating a perfect square in an Algebra 1 problem.

For x²+bx, add (b/2)² to form (x+b/2)²; keep equation sides balanced. To preserve the equation while completing the square.

Worked examples

Try itExample 1.1: Complete the square: x²+8x+__=(x+4)². Find the blank.Show answer

16. Half the x coefficient and square the result. The answer is 16.

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Objective: Explain and use creating a perfect square in an Algebra 1 problem.

For x²+bx, add (b/2)² to form (x+b/2)²; keep equation sides balanced. To preserve the equation while completing the square.

Worked examples

Try itExample 1.1: Complete the square: x²+8x+__=(x+4)². Find the blank.Show answer

16. Half the x coefficient and square the result. The answer is 16.

Objective: Explain and use solving with square roots in an Algebra 1 problem.

After isolating (x−h)²=k, take both positive and negative square roots when k>0. None for a real square.

Worked examples

Try itExample 2.1: Complete the square: x²+4x+__=(x+2)². Find the blank.Show answer

4. Half the x coefficient and square the result. The answer is 4.

Objective: Explain and use vertex form in an Algebra 1 problem.

Writing y=a(x−h)²+k reveals the parabola vertex (h,k) and its vertical opening direction. It opens the parabola downward.

Worked examples

Try itExample 3.1: For y=(x−11)²+37, what is the vertex x-coordinate?Show answer

11. Vertex form reveals the horizontal coordinate where the square is zero. The answer is 11.

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