Solve quadratic equations ax² + bx + c = 0 with the quadratic formula. Find real or complex roots, discriminant Δ, and vertex.
Solve any quadratic equation ax² + bx + c = 0 to get real or complex roots, discriminant (b² - 4ac), and parabola vertex.
Solving Quadratic Equations
A quadratic equation is a second-order polynomial equation in a single variable x written in standard form as:
ax² + bx + c = 0
where a is not equal to 0.
Nature of Roots
Positive Discriminant (Δ > 0): Two distinct real solutions where the parabola crosses the x-axis at two points.
Zero Discriminant (Δ = 0): Exactly one real solution (double root) where the parabola vertex touches the x-axis.
Negative Discriminant (Δ < 0): Two complex conjugate solutions (x = p ± i·q) where the parabola does not cross the x-axis.
How to use
Enter coefficients a, b, and c.
The calculator computes the discriminant Δ and determines root types.
View real roots (x1, x2) or complex conjugate roots, plus the parabola vertex.
Formula
Quadratic Formulax = (-b ± √(b² - 4ac)) / (2a)Formula to find the roots of a quadratic equation.
DiscriminantΔ = b² - 4acDetermines the nature of the roots (real vs complex).
Vertex Formh = -b / (2a), k = c - b² / (4a)The peak or valley point (h, k) of the parabola.
Worked examples
Solving x² - 5x + 6 = 0
For a = 1, b = -5, c = 6: Discriminant Δ = (-5)² - 4(1)(6) = 25 - 24 = 1. Roots are x1 = (5 + 1)/2 = 3, and x2 = (5 - 1)/2 = 2.
Frequently asked questions
What does the discriminant (Δ) tell us?
If Δ is positive, there are two distinct real roots. If Δ is zero, there is one repeated real root. If Δ is negative, there are two complex conjugate roots.
Can coefficient 'a' be zero?
No, if a = 0 the equation becomes linear (bx + c = 0) rather than quadratic.