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Online Quadratic Equation Calculator
Solve any quadratic equation ax² + bx + c = 0 and get the two solutions, the discriminant, and step-by-step working.
| Equation | 1x2 + 8x + 12 = 0 |
|---|---|
| Solution | x = -2 or -6 |
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What Is This Tool?
The Quadratic Equation Calculator solves second-degree equations written in standard form, ax² + bx + c = 0. Enter the coefficients a, b, and c, and it shows the equation it built, the discriminant, and the solutions x₁ and x₂, along with the steps. The solutions are the x-values where the equation equals zero — the points where its parabola crosses the x-axis.
How to Use This Tool?
- Rearrange your equation into the form ax² + bx + c = 0.
- Enter the coefficients a, b, and c.
- Click Calculate to solve the equation.
- Check the equation, discriminant, solutions, and steps.
Key Features
- Solves any quadratic equation in standard form.
- Shows the constructed equation so you can verify your input.
- Displays the discriminant and the step-by-step working.
- Finds two real solutions, one repeated solution, or complex solutions.
- Accepts positive, negative, whole-number, and decimal coefficients.
Examples
- x² + 8x + 12 = 0 has solutions x₁ = -2 and x₂ = -6.
- x² - 8x + 12 = 0 has solutions x₁ = 2 and x₂ = 6.
- Rearranged from 2x² = x + 3, the equation 2x² - x - 3 = 0 gives x = 1.5 and x = -1.
- x² - 4x + 8 = 0 has complex solutions x = 2 ± 2i.
Common Use Cases
- Finding where a parabola crosses the x-axis.
- Solving second-degree equations in algebra and physics.
- Determining the dimensions in area and projectile problems.
- Checking the number and type of solutions from the discriminant.
- Verifying homework solutions with step-by-step working.
Tips & Best Practices
- Move all terms to one side so the equation equals zero before entering values.
- Expand any brackets and combine like terms first.
- Always check the constructed equation matches what you intended.
- Use the discriminant to predict two, one, or no real solutions.
- Remember the coefficient a cannot be zero.
Limitations
- The equation must be rearranged into standard form first.
- The coefficient a cannot be zero, or it is not quadratic.
- Solutions are rounded for readability.
- It solves second-degree equations only, not higher-degree ones.
Frequently Asked Questions
- What is standard form?
- Standard form is ax² + bx + c = 0, with all terms on one side and zero on the other.
- How do I rearrange an equation into standard form?
- Move every term to one side so the other side is zero, expand any brackets, and combine like terms. For example, 2x² = x + 3 becomes 2x² - x - 3 = 0.
- What do x₁ and x₂ mean?
- They are the two solutions of the equation, the x-values where it equals zero and where its parabola crosses the x-axis.
- What if there are no real solutions?
- When the discriminant is negative, the parabola never crosses the x-axis and the solutions are complex, written with i.
Key Terminology
- Quadratic equation
- A second-degree equation of the form ax² + bx + c = 0 where a is not zero.
- Standard form
- A quadratic written with all terms on one side equal to zero.
- Solution (root)
- A value of x that makes the equation true.
- Discriminant
- The value b² - 4ac that reveals the number and type of solutions.
- Parabola
- The U-shaped curve of a quadratic function, whose x-intercepts are the solutions.