Quadratic Equation Calculator
Quadratic Equation Calculation Method
Equation
Discriminant
Quadratic formula
D = 16
x₁ = 2
x₂ = -2
About This Calculator
A quadratic equation ax² + bx + c = 0 is classified by its discriminant D = b² − 4ac: two distinct real roots when D > 0, one repeated root when D = 0, and a complex conjugate pair when D < 0.
For x² − 5x + 6 = 0 the discriminant is 1, so the roots are x = 2 and x = 3. Vieta supplies a useful check: the roots add up to −b/a = 5 and multiply to c/a = 6.
The parabola y = ax² + bx + c has its vertex at x = −b/2a, exactly halfway between the two roots. For x² − 5x + 6 the vertex sits at x = 2.5, midway between the roots 2 and 3, and the minimum value there is −0.25.
Key Formula:
x = (−b ± √(b² − 4ac)) / 2a · D = b² − 4acFrequently Asked Questions
Why does the discriminant matter?
It tells you what kind of roots to expect before you compute them. A positive D means two real roots, zero means a repeated root, and a negative D means no real roots but a complex conjugate pair. On a graph it equals the number of x-axis crossings.
How are complex roots shown?
As a real part plus or minus an imaginary part. For x² + 1 = 0 the discriminant is −4, giving x = 1i and x = −1i. In electrical engineering the imaginary unit is often written as j instead of i.
Where is the vertex of the parabola?
At x = −b/2a. With a = 1 and b = −5 that is x = 2.5. Because the parabola is symmetric about this vertical line, the two roots are always equally far from it, which is a quick way to sanity-check the answers.
How do I factor x² − 5x + 6?
Look for two numbers that add to −b (that is 5) and multiply to c (6): they are 2 and 3, so the expression is (x − 2)(x − 3). Setting each factor to zero gives the same roots the calculator reports.