Equations
Solving quadratics with the quadratic formula worksheets
Solve ten quadratic equations with the quadratic formula. Practice original monic and nonmonic equations with rational and exact radical answers.
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Working with the discriminant worksheets
Calculate the discriminant for twelve quadratic equations, then use its sign and square status to count and classify distinct roots. Choose monic, signed nonmonic or mixed leading coefficients.
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Move every term to the left and identify a, b and c without reducing the equation. Calculate D from those displayed coefficients, then follow the section instruction.
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Customize this worksheetCalculate D before predicting the roots
Each prompt puts the quadratic term on the left and the linear and constant terms on the right. Move them to standard form, retain their signs and calculate D = b² − 4ac from the unreduced coefficients.
A negative D gives two distinct nonreal complex roots. Zero gives one distinct repeated rational root. A positive perfect square gives two distinct rational roots; a positive nonsquare gives two distinct irrational real roots. Every section includes all four cases.
Learning objective
What this activity builds
Identify the standard-form coefficients and use D = b² − 4ac to predict the number and type of distinct roots.
Worked example
For 5k² = −15k + 10, calculate D and classify the roots.
D = 425; two distinct irrational real roots.
Move the terms left: 5k² + 15k − 10 = 0, so a = 5, b = 15 and c = −10. D = 15² − 4(5)(−10) = 425. It is positive and not a perfect square.
Teaching notes
Make the practice count
- Move every term before identifying the coefficients.
- Keep the signs of b and c when substituting.
- Use the displayed coefficients even when they share a common factor.
- Interpret D = 0 as one distinct repeated rational root.
- Distinguish nonreal complex roots from purely imaginary numbers.
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Customize the practice
Choose your options, update the preview, then download the exact same pages.
Each twelve-question sheet has three four-question sections and includes negative, zero, positive-square and positive-nonsquare discriminants in every section.
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