Two real roots
\(x^2 - 5x + 6 = 0 \rightarrow x_1 = 3, x_2 = 2\)
Quadratic Equation Solver
Solve quadratic equations with real and complex solutions
Explore the parabola ax² + bx + c
Drag the a, b, c sliders. Watch the parabola shift, scale, and flip — and how its roots and vertex respond.
Quick presets
1.00
-3.003.00
-5.00
-10.0010.00
6.00
-10.0010.00
Predict what will happen
What happens to the parabola when a becomes negative?
Slide a from 1 to −1 with b = 0, c = 4.
Notice
The discriminant Δ = b² − 4ac decides everything: positive means two real roots, zero means one (a double root), negative means none on the real line.
Common mistake
Don't drop the sign of b in the formula. Use −b, not b. The ± is the square root, not the leading b.
Why it works
The vertex sits at x = −b / (2a) — the axis of symmetry. Real roots are equidistant from the vertex by √Δ / (2a).
Concept check
If the discriminant b² − 4ac is positive, how many real roots does the equation have?
Results
Final Answer
\(x_1 = 3.0000\), \(x_2 = 2.0000\)
Step-by-step Solution
- Given equation: \(1.00x^2 -5.00x + 6.00 = 0\)
- Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
- Calculate discriminant: \(\Delta = -5.00^2 - 4(1.00)(6.00) = 1.00\)
- Two real solutions: \(x_1 = 3.0000\), \(x_2 = 2.0000\)
Discriminant
\(\Delta = 1.00\)
Vertex
\((2.50, -0.25)\)
Theory & Formula
A quadratic equation is a polynomial equation of degree 2, with the general form ax² + bx + c = 0, where a ≠ 0.
The quadratic formula provides the solution(s) to any quadratic equation. The discriminant (b² - 4ac) determines the nature of the roots:
- Δ > 0: Two distinct real roots
- Δ = 0: One real root (double root)
- Δ < 0: Two complex conjugate roots
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Worked Examples
One real root (double)
\(x^2 - 4x + 4 = 0 \rightarrow x = 2\) (double root)
Complex roots
\(x^2 + x + 1 = 0 \rightarrow\) complex
External educational resource
Explore quadratics in PhET
Open PhET's Graphing Quadratics simulation to drag coefficients, see roots and vertex, and verify your intuition.
PhET Interactive Simulations, University of Colorado Boulder