Quadratic Calculator
- Factored form: (3x − 1)(x + 2)
- Vertex form: 3(x + 5/6)² − 49/12
- Opens upward (a is positive), so the vertex is the minimum point.
- Vertex: (−5/6, −49/12), about (−0.8333, −4.0833).
- Axis of symmetry: x = −5/6.
- y-intercept: (0, −2).
- x-intercepts: (1/3, 0) and (−2, 0).
- x = 1/3: 3 × (1/9) + 5 × (1/3) + (−2) = 0.
- x = −2: 3 × 4 + 5 × (−2) + (−2) = 0.
- Sum of roots: −b/a = −5/3. Product of roots: c/a = −2/3.
- Write the equation in standard form: 3x² + 5x − 2 = 0, so a = 3, b = 5 and c = −2.
- Find the discriminant: D = b² − 4ac = 5² − 4 × 3 × (−2) = 49.
- D is positive, so there are two real roots. Apply the quadratic formula: x = (−b ± √D) ÷ 2a.
- x₁ = 1/3 and x₂ = −2.
Quadratic Calculator: Solve Quadratic Equations and Find the Roots
This free quadratic calculator solves any equation of the form ax² + bx + c = 0. Type in the three coefficients, or drag the sliders, and you get the roots, the discriminant, the vertex, the factored form and a graph of the parabola. It handles real roots, repeated roots and complex roots, and it shows the full working so you can follow every step.
Quick answer: for 3x² + 5x − 2 = 0 the discriminant is 5² − 4 × 3 × (−2) = 49, so x = (−5 ± 7) ÷ 6. The two roots are x = 1/3 and x = −2.
What Is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree two, written as ax² + bx + c = 0 with a not equal to zero. Its graph is a U-shaped curve called a parabola. The roots, also called solutions or zeros, are the x-values where the curve crosses the x-axis. Quadratics appear in physics for projectile motion, in business for profit and revenue models, and in geometry whenever an area is involved.
The Quadratic Formula and the Discriminant
The quadratic formula gives both roots at once. The part under the square root, b² − 4ac, is the discriminant, and its sign tells you what to expect before you finish the calculation.
If D is positive there are two real roots, if D is zero there is one repeated root, and if D is negative the roots are complex numbers. The table shows one example of each kind.
| Equation | Discriminant D | Roots |
|---|---|---|
| x² − 5x + 6 = 0 | 1 | x = 2 and x = 3 |
| x² − 4x + 4 = 0 | 0 | x = 2 (repeated root) |
| 3x² + 5x − 2 = 0 | 49 | x = 1/3 and x = −2 |
| 2x² + 3x − 1 = 0 | 17 | x = (−3 ± √17)/4, about 0.2808 and −1.7808 |
| x² + 2x + 5 = 0 | −16 | −1 + 2i and −1 − 2i (complex) |
Vertex, Axis of Symmetry and Factored Form
Every parabola has a turning point called the vertex. Its x-coordinate is −b ÷ 2a, and the vertical line through it is the axis of symmetry. For 3x² + 5x − 2 the vertex is at (−5/6, −49/12), the lowest point of the curve because a is positive. When the roots are rational, the calculator also writes the equation in factored form, here (3x − 1)(x + 2), and always gives the vertex form a(x − h)² + k, which is useful for completing the square.
Check Your Answer with the Sum and Product of Roots
The roots of a quadratic always add up to −b/a and multiply to c/a. For the example above, 1/3 + (−2) = −5/3 and 1/3 × (−2) = −2/3, which match. The calculator shows this check on every result, along with a direct substitution when the roots are rational. If your equation has two unknowns instead, try the linear equations calculator.
Who Uses a Quadratic Calculator?
Students use it to check homework and to see how a result is reached, teachers use it to build examples, and engineers, designers and analysts use it when a model has a squared term. For decimals or trigonometry alongside your algebra, the scientific calculator is a good partner, and the fraction calculator helps when you want to simplify an exact answer by hand.
How to Use This Quadratic Calculator
Write the equation in standard form
Move everything to one side so it reads ax² + bx + c = 0.
Enter a, b and c
Type each value or drag its slider. Use a minus sign for negative numbers and 0 when a term is missing.
Check the equation preview
The equation is shown above the boxes as you type, so you can see it is entered correctly.
Read the roots and the working
The roots, discriminant, vertex, graph, check and steps update as soon as you change a number.