havo vwo
Quadratic formula calculator
Quick answer
For ax² + bx + c = 0, D = b² − 4ac and x = (−b ± √D) / (2a). Dutch schools call this the abc-formule (quadratic formula).
Example: x² − 5x + 6 = 0 gives D = 1, so x = 2 or x = 3. If a = 0, use the linear equation solver.
Calculator
Enter a, b and c. On iOS the decimal keypad has no minus key: use ± next to the field.
What is the quadratic formula?
A quadratic equation has the shape ax² + bx + c = 0 with a ≠ 0. The quadratic formula finds the solutions through the discriminant D = b² − 4ac, then x = (−b ± √D) / (2a). In Dutch secondary school the same method is taught as the abc-formule — that is simply the Dutch school name for the quadratic formula.
| D | Real solutions | Example |
|---|---|---|
| D > 0 | Two different | x² − 5x + 6 = 0 → D = 1 → x = 2 or x = 3 |
| D = 0 | One (repeated root) | x² + 2x + 1 = 0 → D = 0 → x = −1 |
| D < 0 | None real | x² + x + 1 = 0 → D = −3 |
Complex solutions are not part of the havo program; on vwo they appear later, not in this tool. If a = 0 the x² term disappears — use linear equation solver.
Sometimes you can factor first: (x − 2)(x − 3) = 0 gives x = 2 or x = 3 at once. The quadratic formula is the general method when factoring does not work.
Examples by level
The tabs use Dutch school names once: vmbo is the more practical track, havo sits in the middle, vwo is the university-preparatory track. The quadratic formula is a regular topic on havo and vwo, not a fixed part of the vmbo-bb/kb central exam program — so there is no vmbo tab here.
Solve: x² − 5x + 6 = 0.
- a, b, ca = 1, b = −5, c = 6
- DiscriminantD = (−5)² − 4·1·6 = 25 − 24 = 1
- Rootsx = (5 ± 1) / 2 → x = 3 or x = 2
Check by factoring: (x − 2)(x − 3) = x² − 5x + 6. Watch the sign: −b is −(−5) = +5.
2x² − 4x − 6 = 0. Simplify first, then use the formula.
- Divide by 2x² − 2x − 3 = 0 (same roots)
- D(−2)² − 4·1·(−3) = 4 + 12 = 16
- x(2 ± 4) / 2 → x = 3 or x = −1
Without dividing: D = (−4)² − 4·2·(−6) = 16 + 48 = 64, x = (4 ± 8) / 4, same answers.
Common mistakes
Wrong
D = b² − 4c, forgetting a.
Right
D = b² − 4ac. All three letters appear in the discriminant.
Wrong
With b = −5 writing −b in the numerator as −5 instead of +5.
Right
−b = −(−5) = 5. Then (5 ± √D) / (2a).
Wrong
For x² + 2x + 1 = 0 writing two different answers.
Right
D = 0, so one solution: x = −1 (repeated root).
Wrong
When D < 0 forcing √D on the calculator and writing down a “answer”.
Right
D < 0: no real solutions. Stop there; this tool does not return complex roots.
Practice
havo x² + 2x + 1 = 0. How many real solutions?
One: x = −1
D = 4 − 4 = 0.
havo x² + x + 1 = 0. What is D?
D = −3, no real solutions
1² − 4·1·1 = 1 − 4 = −3.
vwo 2x² − 4x − 6 = 0. What are the roots?
x = 3 or x = −1
After dividing by 2: x² − 2x − 3 = 0, D = 16.
vwo Why may a not be 0?
Then x² disappears. The equation is linear; dividing by 2a in the formula would be dividing by 0. Open linear equation solver.
On havo and vwo
Quadratic equations belong to mathematics A and B on havo and vwo. On vmbo-bb/kb the quadratic formula is not a fixed part of the central exam program; that is why there is no vmbo tab above. See Examenblad.nl for the program that applies to your track.
Next step
- How does this work? The explanation and examples are further up this page.
- See it visually Parabola simulation: change the values and watch what happens.
- Test yourself Quadratic formula practice, with an explanation after every answer.
- Video explanations On the Parabola page. YouTube only loads when you press play.
Learning path
- Related: Linear equation solver · Systems of equations
Frequently asked questions
What is the quadratic formula?
First D = b² − 4ac, then x = (−b ± √D) / (2a). In Dutch schools this is called the abc-formule.
What does the discriminant tell you?
D > 0 two real solutions, D = 0 one real solution, D < 0 no real solutions.
What if D < 0?
No real roots. Example: x² + x + 1 = 0 has D = −3.
What if a is 0?
Then the equation is linear. Open the linear equation solver.