Quick Answer: Enter the coefficients a, b, and c from your equation ax² + bx + c = 0 and press Solve with Steps. The calculator instantly returns both roots (real or complex) together with the full step-by-step working – the formula, the substitution, the discriminant, and each root calculated line by line, exactly the way you would show it in an exam.
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation of the form:
ax² + bx + c = 0, where a ≠ 0
- a = coefficient of x² (must not be zero, otherwise the equation is linear)
- b = coefficient of x
- c = constant term
Its graph is a parabola, and the solutions (roots) are the x-values where the parabola crosses the x-axis. Example: 2x² + 3x – 5 = 0 has a = 2, b = 3, c = –5.
The Quadratic Formula
The quadratic formula solves any quadratic equation, whether or not it can be factored:
x = (–b ± √(b² – 4ac)) / 2a
The expression under the square root, b² – 4ac, is called the discriminant (D). It tells you what kind of roots to expect before you solve anything:
| Discriminant | Nature of Roots | Graph Meaning |
|---|---|---|
| D > 0 | Two distinct real roots | Parabola crosses x-axis at 2 points |
| D = 0 | One repeated real root | Parabola touches x-axis at 1 point |
| D < 0 | Two complex conjugate roots | Parabola never touches x-axis |
Worked Example: Solve x² – 5x + 6 = 0 Step by Step
- Identify coefficients: a = 1, b = –5, c = 6
- Compute the discriminant: D = (–5)² – 4(1)(6) = 25 – 24 = 1
- Square root: √1 = 1
- Apply the formula: x = (5 ± 1) / 2
- Both roots: x₁ = 6/2 = 3 and x₂ = 4/2 = 2
Check: 3² – 5(3) + 6 = 9 – 15 + 6 = 0 ✓ You can verify any answer by substituting the root back into the original equation.
Example with Complex Roots: x² + 2x + 5 = 0
Here a = 1, b = 2, c = 5, so D = 4 – 20 = –16. Since the discriminant is negative, there are no real solutions. Taking √16 = 4 as the imaginary part: x = (–2 ± 4i) / 2, which gives x₁ = –1 + 2i and x₂ = –1 – 2i. The calculator above handles complex roots automatically.
How to Use the Quadratic Formula Calculator
- Enter the values of a, b, and c from your equation (negative and decimal values are supported).
- Press Solve with Steps.
- Read both roots in the green solution box.
- Follow the blue step-by-step solution to see the formula, substitution, discriminant, and each root worked out – ideal for checking homework.
Why Use This Quadratic Equation Solver?
- Shows full working – not just the answer, but every step, so you learn the method.
- Handles every case – two real roots, repeated roots, complex roots, even accidental linear equations (a = 0).
- Students – verify homework and exam practice instantly.
- Teachers – demonstrate the discriminant and root cases live in class.
- Engineers and scientists – quick root-finding for physics, projectile motion, and economics problems.
Frequently Asked Questions
What is the quadratic formula used for?
The quadratic formula is used to solve quadratic equations of the form ax² + bx + c = 0, finding the exact value(s) of x that make the equation true. It works for every quadratic equation, even ones that cannot be factored.
Does this calculator show step-by-step solutions?
Yes. After you press Solve with Steps, it shows the formula, the substitution of your values, the discriminant calculation, the square root, and both roots worked out line by line – the same working you would write by hand.
Can it solve equations with imaginary or complex roots?
Yes. When the discriminant (b² – 4ac) is negative, the calculator automatically returns the two complex conjugate roots in the form a + bi and a – bi.
What does it mean if the discriminant is zero?
A discriminant of exactly zero means the equation has one repeated real root – the parabola touches the x-axis at exactly one point instead of crossing it twice.
Can I enter decimal or negative coefficients?
Yes. The calculator accepts any real numbers for a, b, and c, including decimals like 0.5 and negatives like –3, and rounds results to six decimal places.
Is this quadratic equation solver free to use?
Yes, it is completely free with no sign-up required, and you can solve as many equations as you like.