Quadratic Formula Calculator
Solve ax² + bx + c = 0 for x, including cases with complex roots.
How this calculator works
The quadratic formula solves any equation of the form ax² + bx + c = 0. When the discriminant is negative, the roots are complex numbers.
a, b, c are the coefficients of the quadratic equation.
Understanding the discriminant
The discriminant (b² − 4ac) tells you what type of roots to expect before even solving the full equation. A positive discriminant means two distinct real roots, zero means exactly one repeated real root, and a negative discriminant means two complex (non-real) roots.
This is useful for quickly checking whether a quadratic equation has real solutions at all — in applied contexts (like physics problems), a negative discriminant often signals that the scenario as posed has no real physical solution.
A worked example
For x squared minus 5x plus 6 equals 0 (a=1, b=-5, c=6): the discriminant is 25 minus 24, or 1 (positive), giving two real roots: x equals (5 plus or minus 1) divided by 2, so x equals 3 or x equals 2.
For x squared plus 2x plus 5 equals 0 (a=1, b=2, c=5): the discriminant is 4 minus 20, or negative 16, giving complex roots: x equals negative 1 plus or minus 2i, since there's no real number solution to this equation.
Interpreting the discriminant
The discriminant (b² − 4ac) reveals the nature of a quadratic equation's roots before you even solve for them: a positive discriminant means two distinct real roots, zero means exactly one repeated real root (the parabola touches the x-axis at a single point), and a negative discriminant means two complex roots (the parabola never crosses the x-axis).
This connects directly to graphing — a quadratic equation's roots are the x-intercepts of its parabola, so the discriminant essentially tells you how many times (if any) the parabola crosses the x-axis.
Frequently asked questions
What if the discriminant is negative?
The equation has two complex (non-real) roots. This calculator shows them in the form a ± bi.
What if a = 0?
The equation is no longer quadratic (it becomes linear), so the quadratic formula doesn't apply — solve bx + c = 0 directly instead.
What does it mean if the roots are complex numbers?
It means the equation has no real number solutions — graphically, the parabola never crosses the x-axis. Complex roots are still mathematically valid but represent no real x-intercept.
Can I use this for equations not already in ax² + bx + c = 0 form?
Rearrange your equation into that standard form first — move all terms to one side and set equal to zero, then identify your a, b, and c coefficients.
What does it mean if the discriminant is zero?
It means the quadratic has exactly one repeated real root — graphically, the parabola's vertex touches the x-axis at exactly one point rather than crossing it.
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b²-4ac)) / 2a. It finds the values of x that solve any equation in the form ax² + bx + c = 0. The ± means there are usually two solutions (two roots).
What is the discriminant in the quadratic formula?
The discriminant is b² - 4ac (the part under the square root). If it's positive, the equation has two real roots. If it equals zero, there is exactly one real root. If it's negative, the roots are complex (imaginary) numbers.
How do you use the quadratic formula step by step?
Step 1: Write the equation in standard form ax² + bx + c = 0. Step 2: Identify a, b, and c. Step 3: Calculate the discriminant: b² - 4ac. Step 4: Apply the formula: x = (-b ± √(b²-4ac)) / 2a. Step 5: Solve for both + and - to get two roots.