Completing the Square

Algebra Foundation

Algebra 1

Study expressions, equations, inequalities, functions, linear models, systems, polynomials, quadratics, and exponential growth with original explanations and worked examples.

Learn with original explanations and worked examples. OpenStax Elementary Algebra 2e was consulted for topic coverage.

Study every current unit through original concept lessons, formula derivations, experiments, quizzes, and practice sets.

Algebra 1 · Unit 9 · Topic 9.3

Completing the Square

Completing the square rewrites x² + bx using (x + b/2)² − (b/2)².

Definition and explanation

Completing the square rewrites x² + bx using (x + b/2)² − (b/2)². In an equation, add the same quantity to both sides; for a nonunit leading coefficient, account for it first.

Worked example

Solve x² + 6x = 7. Add 9 to both sides: (x + 3)² = 16. Hence x + 3 = ±4 and x = 1 or −7. Both satisfy the original equation.

Common mistake

The number to complete the square is (b/2)², not b² or b/2.

Key takeaway

Completing the square rewrites x² + bx using (x + b/2)² − (b/2)².

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Elementary Algebra 2e and Algebra and Trigonometry 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.