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Quadratic Equation Solver
Solve equations in the form: ax² + bx + c = 0
a = 2, b = -5, c = 3
Results
Quadratic Formula & Concepts
The Quadratic Formula
This formula finds the values of x where ax² + bx + c = 0
What is the Discriminant?
The discriminant (Δ) tells us about the nature of the roots:
- Δ > 0: Two distinct real roots
- Δ = 0: One repeated real root (double root)
- Δ < 0: Two complex conjugate roots (no real solutions)
Vertex of a Parabola
Axis of Symmetry
This is the vertical line around which the parabola is symmetric.
Key Concepts
- Quadratic Equation: An equation of degree 2 in the form ax² + bx + c = 0
- Root/Zero: A value of x that makes the equation equal to zero
- Parabola: The graph of a quadratic function (U-shaped)
- Real Roots: Roots that are real numbers (not complex)
- Complex Roots: Roots that contain imaginary numbers (involve √-1)
- Vertex: The highest or lowest point of the parabola
How to Use the Calculator
Understanding Your Results
- Discriminant: Shows the value of b² - 4ac
- Root Type: Tells you if roots are real or complex
- Root 1 & Root 2: The solutions to your equation
- Vertex: The turning point of the parabola
- Axis of Symmetry: The line where the parabola mirrors itself
Understanding Quadratic Equations
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree 2. It has the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
Real-World Applications
- Physics: Projectile motion, falling objects, and trajectory calculations
- Engineering: Structural analysis and design problems
- Economics: Profit and loss calculations, cost optimization
- Business: Revenue and pricing models
- Medicine: Drug concentration and dosage calculations
- Construction: Area calculations and design parameters
Example Problems
- Problem: x² - 5x + 6 = 0
Solution: a = 1, b = -5, c = 6 → x = 2 or x = 3 - Problem: 2x² - 4x + 2 = 0
Solution: a = 2, b = -4, c = 2 → x = 1 (double root) - Problem: x² + 2x + 5 = 0
Solution: a = 1, b = 2, c = 5 → Complex roots (no real solutions)
Frequently Asked Questions
Can a be zero?
No, a cannot be zero. If a = 0, the equation becomes linear (bx + c = 0), not quadratic. You must have a ≠ 0.
What if the discriminant is negative?
A negative discriminant means there are no real solutions, only complex roots. The parabola doesn't cross the x-axis.
What is a double root?
A double root occurs when the discriminant equals zero. Both roots are the same value. The parabola touches the x-axis at exactly one point.
How do I find the vertex?
The x-coordinate of the vertex is h = -b/2a. Substitute this into the equation to find the y-coordinate (k).
What's the axis of symmetry?
It's a vertical line at x = -b/2a. The parabola is symmetric (mirrored) about this line.
How do I factor a quadratic?
Factoring writes the equation as a(x - r₁)(x - r₂) = 0, where r₁ and r₂ are the roots. You can find the roots using the quadratic formula.
What if I get complex roots?
Complex roots have imaginary parts (involving i = √-1). This means the parabola doesn't cross the real x-axis.
Can b or c be zero?
Yes! Both b and c can be zero. You can have equations like x² - 4 = 0 or x² - 5x = 0. Only a cannot be zero.
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