The worksheets themselves are in German – just as in class. The editor is only available in German so far.
Quadratic equations and functions – worksheets to print
Solve quadratic equations – by taking square roots, by factoring out x, with the quadratic formula and with Vieta's formulas – and explore parabolas: the vertex from the vertex form, by completing the square and from the graph.
- Choose the number of tasks and the difficulty yourself
- With an answer sheet and, if you like, a QR code to the answers
- At three levels (★ to ★★★) and as group A and B
- As a single worksheet or as a task on a mixed worksheet
Quadratic equations and functions: explained simply
A quadratic equation contains x² – it has two solutions, one or none. The graph of the matching function y = ax² + bx + c is a parabola with a vertex.
How to work it out
- Only x² (x² = 49): take the square root and remember both signs: x₁ = −7, x₂ = 7. No constant term (x² − 5x = 0): factor out x, x · (x − 5) = 0, so x₁ = 0, x₂ = 5.
- The pq formula for x² + px + q = 0: x = −p/2 ± √((p/2)² − q). If the number under the root (the discriminant) is negative, there is no solution; if it is 0, there is exactly one.
- Vertex form y = a(x − d)² + e: the vertex is S(d | e). If a is positive, the parabola opens upwards, otherwise downwards.
An example, step by step
Solve x² − 2x − 15 = 0.
- p = −2 and q = −15, so p/2 = −1.
- x = 1 ± √(1 + 15) = 1 ± 4.
- x₁ = −3 and x₂ = 5. Check: 25 − 10 − 15 = 0.
Understand other variations
Vieta's formulas
For x² + px + q = 0: x₁ + x₂ = −p and x₁ · x₂ = q. For x² − 2x − 15 = 0: −3 + 5 = 2 and −3 · 5 = −15. This also gives the factored form (x + 3)(x − 5) = 0.
The quadratic formula
For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) : (2a). For 2x² + 5x − 12 = 0: x = (−5 ± √121) : 4, so x₁ = −4 and x₂ = 1.5.
Completing the square
y = x² − 6x + 11 = x² − 6x + 9 − 9 + 11 = (x − 3)² + 2. The vertex is S(3 | 2).
Shifting the parabola
y = (x − 3)² + 2 is the standard parabola moved 3 to the right and 2 up. Careful: (x + 3)² means 3 to the left.
Your turn
Solve x² + 4x − 12 = 0.
Show the answer and method
p/2 = 2: x = −2 ± √(4 + 12) = −2 ± 4, so x₁ = −6 and x₂ = 2. Vieta: −6 + 2 = −4 = −p and −6 · 2 = −12 = q.
Ready-made worksheets: Quadratic equations and functions
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