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Quadratic Equations Worksheets

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Easy

10 problems

Medium

20 problems

Hard

20 problems

Mixed

30 problems

Free printable quadratic equations worksheets with step-by-step answer keys. Every worksheet is uniquely generated so students never see the same problems twice. Topics covered range from perfect square equations at the easy level through to factoring, larger root range at the advanced level.

CCSS.HSA.REICCSS.HSA.SSE

What is quadratic equations?

A quadratic equation is a polynomial equation of degree 2, written in standard form as ax² + bx + c = 0, where a ≠ 0. These equations have at most 2 solutions, which represent the x-intercepts of a parabola when graphed. The solutions can be found through factoring, completing the square, or the quadratic formula.

Why it matters

Quadratic equations model countless real-world scenarios involving area, projectile motion, and optimization. A baseball's height follows the equation h = -16t² + 64t + 5, where solutions determine when it hits the ground. In business, profit functions like P = -2x² + 100x - 800 help companies find break-even points at x = 10 and x = 40 units. Engineers use quadratics to design parabolic satellite dishes and suspension bridge cables. The Pythagorean theorem often leads to quadratics when finding unknown side lengths. Quadratics appear throughout algebra, precalculus, and calculus, forming the foundation for polynomial functions, conic sections, and optimization problems in advanced mathematics.

Common mistakes to watch for

  • ✗When factoring x² - 25 = 0, writing x = 5 instead of recognizing both solutions x = 5 and x = -5 from the difference of squares pattern
  • ✗In x² + 6x + 9 = 0, incorrectly factoring as (x + 3)(x + 3) = 0 but concluding there are two different solutions instead of one repeated solution x = -3
  • ✗Using the quadratic formula on 2x² - 8x + 6 = 0 and getting x = (8 ± √16)/4 = 3 or 1, forgetting that the discriminant calculation should be b² - 4ac = 64 - 48 = 16

Questions teachers ask

What is the difference between linear and quadratic equations?+
Linear equations have degree 1 (highest power of x is 1) and graph as straight lines with exactly one solution. Quadratic equations have degree 2 (highest power is x²) and graph as parabolas with 0, 1, or 2 solutions. For example, 2x + 5 = 0 has one solution x = -2.5, while x² - 4 = 0 has two solutions x = 2 and x = -2.
How do you know when a quadratic equation has no real solutions?+
A quadratic equation has no real solutions when the discriminant (b² - 4ac) is negative. For example, in x² + 2x + 5 = 0, the discriminant equals 4 - 20 = -16. Since this is negative, the parabola never crosses the x-axis and the equation has no real solutions, only complex ones.
When should you use factoring versus the quadratic formula?+
Use factoring when the quadratic has integer solutions and factors easily, like x² - 7x + 12 = (x - 3)(x - 4). Use the quadratic formula for equations that don't factor nicely, such as 2x² + 3x - 1 = 0. The quadratic formula always works, but factoring is faster when possible.
What does it mean when a quadratic has one solution?+
One solution occurs when the discriminant equals zero, creating a perfect square trinomial. For example, x² - 6x + 9 = 0 factors to (x - 3)² = 0, giving the repeated solution x = 3. Graphically, this means the parabola touches the x-axis at exactly one point (the vertex).
How do you check if your quadratic equation solutions are correct?+
Substitute each solution back into the original equation. For x² - 5x + 6 = 0 with solutions x = 2 and x = 3, check: (2)² - 5(2) + 6 = 4 - 10 + 6 = 0 ✓ and (3)² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓. Both should equal zero.
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Pick a difficulty

Click any level to open the generator with that difficulty pre-selected.

Beginner

Generate →
Concepts
Perfect square equations
Range
x: 1–5
Steps
2–3 steps
Example
x² = 25

Easy

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Concepts
Factoring with positive integer roots
Range
roots 1–5, a=1
Steps
4–5 steps
Example
x² − 5x + 6 = 0

Medium

Generate →
Concepts
Factoring, roots may be negative
Range
roots −6 to 6, a=1
Steps
4–5 steps
Example
x² + x − 12 = 0

Hard

Generate →
Concepts
Factoring, larger root range
Range
roots −8 to 8, a=1
Steps
4–5 steps
Example
x² − 2x − 15 = 0

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