Incomplete quadratics
For ax² + c = 0, make x² the subject and take the square root (two opposite solutions or none); solve ax² + bx = 0 by taking out x, never by dividing by x.
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Teaching Guide: Solving Quadratic Equations
Quadratic equations are the big step up in algebra in Grade 9: from incomplete quadratics, solved by making x² the subject or taking out x, to the quadratic formula and the discriminant. This free quadratic equations worksheet generator builds A4 worksheets of equations made from their solutions, with an answer key that shows every step.
Student worksheet 
Answer key The quadratic formula
Identify a, b and c, work out b² − 4ac with brackets around negative numbers and use x = (−b ± √(b² − 4ac))/2a to reach both solutions in their simplest form.
Expand and rearrange
Products, squared brackets such as (x + 3)², x on both sides and fractions: everything is brought to ax² + bx + c = 0 before solving.
The discriminant
Two solutions, one repeated root or none depending on the sign of b² − 4ac, without solving. In Grade 10, the value of k that gives a repeated root.
Sum and product
From the roots to the equation with x² − Sx + P = 0, including fraction roots, and the classic puzzle of two numbers with a given sum and product.
Step-by-step answer key
Every step named with its result: clear the fractions, expand, collect terms, simplify, the discriminant, the formula and the solutions highlighted.
Frequently Asked Questions about the Quadratic Equations
Do the solutions come out exactly?
Yes. Each equation is built from its solutions (integers or fractions) and the generator solves it again step by step with exact arithmetic, so the discriminant is always a perfect square when the equation has solutions.
Which school years are they for?
Incomplete quadratics, the quadratic formula and the discriminant suit Grade 9; equations with fractions, the parameter k and the sum and product problems fit Grade 10. Students should be confident with linear equations and fraction operations first.
Why does the answer key sometimes divide the equation first?
When all the coefficients share a factor, or the coefficient of x² is negative, dividing first (or changing the signs) keeps the numbers small and the formula simpler. The solutions are the same.
Can I mix several types on one worksheet?
Yes. Add one exercise for each type and level you want, or from other generators such as order of operations, and they all share the worksheet and the answer key. Save it under Templates to use it again.
Is any student data sent to a server?
No. Every equation is generated in your browser and no student data is stored.
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