Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 instantly. Find real and complex roots, calculate the discriminant, vertex, and axis of symmetry with step-by-step solutions.
Quadratic Equation Solver
Enter the coefficients for ax² + bx + c = 0
Cannot be zero
Equation Solutions
Roots, discriminant, and parabola properties
Enter your coefficients and click Solve Equation to see the roots and step-by-step breakdown.
Common Quadratic Equations
Examples of quadratic equations, their discriminants, and solutions.
| Equation | a, b, c | Discriminant (Δ) | Roots (Solutions) |
|---|---|---|---|
| x² - 5x + 6 = 0 | 1, -5, 6 | 1 (Positive) | x = 2, x = 3 |
| x² + 4x + 4 = 0 | 1, 4, 4 | 0 (Zero) | x = -2 (Repeated) |
| x² + x + 1 = 0 | 1, 1, 1 | -3 (Negative) | Complex roots |
| 2x² - 8 = 0 | 2, 0, -8 | 64 (Positive) | x = 2, x = -2 |
| x² - 7x + 10 = 0 | 1, -7, 10 | 9 (Positive) | x = 2, x = 5 |
| 3x² + 6x + 3 = 0 | 3, 6, 3 | 0 (Zero) | x = -1 (Repeated) |
Quadratic Formula FAQ
Everything you need to know about solving quadratic equations and understanding parabolas.
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It provides the solutions (roots) to any quadratic equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It is derived by completing the square on the general quadratic equation.
The discriminant is the expression under the square root in the quadratic formula: Δ = b² - 4ac. It determines the nature of the roots: if Δ > 0, there are two distinct real roots; if Δ = 0, there is one repeated real root; and if Δ < 0, there are two complex (imaginary) roots. It tells you how many x-intercepts the parabola has.
To solve a quadratic equation ax² + bx + c = 0: Step 1: Ensure the equation is in standard form and identify the coefficients a, b, and c. Step 2: Calculate the discriminant (b² - 4ac). Step 3: Substitute a, b, and c into the quadratic formula. Step 4: Simplify the expression to find the values of x. This calculator performs all these steps automatically and shows the work.
Yes, a quadratic equation can have no real solutions. This happens when the discriminant (b² - 4ac) is negative. In this case, the square root of a negative number results in imaginary numbers, meaning the parabola does not cross the x-axis. The solutions are expressed as complex numbers (e.g., a ± bi).
The vertex is the highest or lowest point of a parabola, representing its maximum or minimum value. The x-coordinate of the vertex is found using the formula h = -b / 2a. To find the y-coordinate (k), substitute h back into the original equation: k = f(h) = a(h)² + b(h) + c. The vertex form of a quadratic is y = a(x - h)² + k.
The axis of symmetry is a vertical line that divides the parabola into two identical halves. It always passes through the vertex. The formula for the axis of symmetry is x = -b / 2a. For example, in the equation x² - 4x + 3 = 0, the axis of symmetry is x = -(-4) / 2(1) = 2.
If 'a' is zero, the equation is no longer quadratic; it becomes a linear equation (bx + c = 0). The quadratic formula would involve division by zero (2a = 0), which is undefined. In this case, you simply solve the linear equation by isolating x: x = -c / b.
The term "quadratic" comes from the Latin word "quadratus", meaning "square". It is called the quadratic formula because it solves quadratic equations, which are polynomials of degree 2 — meaning the highest power of the variable x is squared (x²). Other degrees have different names: degree 1 is linear, degree 3 is cubic, and degree 4 is quartic.