Polynomials
Completing the square worksheets
Find the missing constant in ten monic quadratic trinomials. Choose whole, fractional or mixed linear coefficients and check exact reduced answers.
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Solving quadratics by completing the square worksheets
Solve ten quadratic equations by completing the square. Choose easy, hard or mixed equations and rational-only answers or exact radical roots.
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Start with a ready-made worksheet
Move the equation to standard form, complete the square and give every exact real root.
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Customize this worksheetUse an exact quadratic method and keep every real root
Move every term to the left before applying the selected method. Completing the square rewrites the expression as a squared binomial; the quadratic formula substitutes the displayed standard-form coefficients. Both routes preserve exact rational and radical roots.
Each sheet balances five monic and five nonmonic displays and includes one repeated root. Radical practice includes four irrational conjugate pairs. Keys show the reduced root set and one compact cue using the original displayed equation.
Learning objective
What this activity builds
Rewrite a quadratic as a squared binomial and solve its exact real root set.
Worked example
Solve x² − 14x − 24 = 0.
x = 7 ± √73.
Move the constant and add 49 to both sides: (x − 7)² = 73. Taking both square roots gives x − 7 = ±√73.
Teaching notes
Make the practice count
- Divide by the leading coefficient before completing a nonmonic square.
- Keep fractional shifts exact.
- Use both signs after taking a square root.
- Write a repeated root once.
Make it yours
Customize the practice
Choose your options, update the preview, then download the exact same pages.
Every sheet balances monic and nonmonic equations, includes one repeated root and keeps rational or radical answers exact.
Enlarged pages scroll. PDF size stays the same.
A useful next step
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Solve ten factorable quadratic equations with complete exact root sets. Choose factored form, expanded form, rearrangement practice or a balanced mixed sheet.
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