Quadratic Equation Solver
Solve quadratic equations in standard form: ax² + bx + c = 0
Equation: ax² + bx + c = 0
Must not be zero
Your Equation:
1x² -7x +10 = 0
Solutions
Two distinct real solutions
x₁ =
5.0000
x₂ =
2.0000
Discriminant (Δ)
9.0000
b² - 4ac = -7² - 4(1)(10)
Vertex
(3.50, -2.25)
Parabola minimum point
Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
Factored Form
(x - 5.00)(x - 2.00) = 0
Understanding Quadratic Equations
A quadratic equation is a second-degree polynomial equation in standard form: ax² + bx + c = 0, where a ≠ 0. The solutions represent where the parabola crosses the x-axis.
Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
Discriminant (Δ = b² - 4ac)
- Δ > 0: Two distinct real roots (parabola crosses x-axis twice)
- Δ = 0: One repeated real root (parabola touches x-axis once)
- Δ < 0: No real roots (parabola doesn't cross x-axis)
Example
Solve: x² - 7x + 10 = 0
- a = 1, b = -7, c = 10
- Δ = (-7)² - 4(1)(10) = 49 - 40 = 9
- x = (7 ± √9) / 2 = (7 ± 3) / 2
- x₁ = 5, x₂ = 2
- Check: (x-5)(x-2) = x² - 7x + 10 ✓
Applications
- Physics: Projectile motion, free fall
- Engineering: Optimization problems
- Business: Profit/loss analysis
- Geometry: Area and perimeter problems