Quadratic equation solver
The quadratic formula, worked through to the exact answer.
Output
Paste or type above to see the result here.
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Example
Example: a garden bed 4 m longer than it is wide, with an area of 21 m²
A rectangular bed is 4 m longer than it is wide and covers 21 m². With x for the width, x(x + 4) = 21, which is x² + 4x − 21 = 0. Type that line — or just the three numbers 1 4 -21.
- a = 1, b = 4, c = −21, so the discriminant is b² − 4ac = 4² − 4·1·(−21) = 100.
- The formula gives x = (−4 ± √100) / 2 = (−4 ± 10) / 2, so x₁ = −7 and x₂ = 3.
- Then the factored form (x + 7)(x − 3) = 0, the vertex (−2, −25) and the check: −7 + 3 = −4 = −b/a and −7 · 3 = −21 = c/a.
| Equation | Discriminant | Solutions |
|---|---|---|
| x² + 4x − 21 = 0 | 100 | x₁ = −7, x₂ = 3 |
| 2x² − 7x + 3 = 0 | 25 | x₁ = 1/2, x₂ = 3 |
| x² − 4x + 1 = 0 | 12 | x = 2 ± √3 (0.267949 and 3.732051) |
| x² + 2x + 5 = 0 | −16 | no real solution; x = −1 ± 2i |
A width cannot be negative, so the bed is 3 m wide and 7 m long: 3 × 7 = 21 m². The solver gives both roots; which one fits is for the problem to say.
How it works
- Type the equation: x² − 5x + 6 = 0, 2x^2 + 3x = 2 (terms can be on both sides, and fractions or decimals are fine) — or just the three numbers a b c.
- Put each equation on its own line to solve several at once.
- The solutions appear first, exact where possible, then the working: standard form, discriminant, formula, factored form, vertex and check.
Frequently asked questions
- What is the quadratic formula?
- For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / (2a). The part under the root, b² − 4ac, is the discriminant.
- What does the discriminant tell you?
- If it is positive there are two real solutions, if it is zero one (a double root), if it is negative none — only a pair of complex solutions, which the solver shows too.
- Why is the answer 2 ± √3 and not a decimal?
- Because that is exact. The root is simplified (√12 = 2√3) and the decimals are given beside it: 0.267949 and 3.732051.
- What if a is 0?
- Then it is not a quadratic but a linear equation, and the solver solves it as one.
- How do I factor a quadratic?
- When the solutions are whole numbers or fractions, the solver writes the factored form for you: x² + 4x − 21 = (x + 7)(x − 3), and 2x² − 7x + 3 = 2(x − 1/2)(x − 3). With solutions like 2 ± √3 there are no such factors, and none is shown.
- How do I find the vertex?
- It is shown with the working: at x = −b/2a, with the value of the expression there. For x² + 4x − 21 the vertex is (−2, −25), the lowest point of the parabola, since a is positive.
- Is anything uploaded?
- No. It is worked out in your browser, and nothing you type leaves your computer.
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