Quadratic Equation Solver
Enter Coefficients
For equation: ax² + bx + c = 0
Solution Results
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A quadratic equation is a polynomial equation of degree 2. The discriminant (b² - 4ac) determines the nature of roots: positive = 2 real roots, zero = 1 real root, negative = 2 complex roots.
Complete Quadratic Equation Guide
Quadratic equations are fundamental in algebra and appear in many real-world applications. This guide explains how to solve them and interpret the results.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree 2, written as ax² + bx + c = 0, where a ≠ 0. The solutions to this equation are called roots or zeros, representing the values of x that satisfy the equation.
The Quadratic Formula
x = [-b ± √(b² - 4ac)] / (2a)
This formula provides the roots directly by substituting the coefficients a, b, and c.
The Discriminant
| Discriminant (Δ = b² - 4ac) | Nature of Roots | Number of Roots | Type |
|---|---|---|---|
| Δ > 0 | Real and Unequal | 2 distinct real roots | Parabola crosses x-axis twice |
| Δ = 0 | Real and Equal | 1 repeated real root | Parabola touches x-axis once |
| Δ < 0 | Complex/Imaginary | 2 complex conjugate roots | Parabola doesn't touch x-axis |
Key Properties of Quadratic Equations
- Vertex Form: y = a(x - h)² + k, where (h, k) is the vertex
- Vertex Coordinates: x = -b/(2a), y = c - b²/(4a)
- Axis of Symmetry: x = -b/(2a) (vertical line through vertex)
- Y-intercept: When x = 0, y = c
- X-intercepts (roots): Where the parabola crosses the x-axis
- Direction: Opens upward if a > 0, downward if a < 0
Methods to Solve Quadratic Equations
- Quadratic Formula: Direct method using the formula. Works for all equations.
- Factoring: Express as (x - r₁)(x - r₂) = 0. Works when roots are rational.
- Completing the Square: Rewrite as a perfect square trinomial.
- Graphing: Plot the parabola and find x-intercepts visually.
- Vieta's Formulas: Sum of roots = -b/a, Product of roots = c/a
Practical Examples
Equation: x² - 5x + 6 = 0 (a=1, b=-5, c=6)
Discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1 > 0
Roots: x = [5 ± 1] / 2 = 3 or 2
Factored form: (x - 2)(x - 3) = 0
Equation: x² - 4x + 4 = 0 (a=1, b=-4, c=4)
Discriminant: (-4)² - 4(1)(4) = 16 - 16 = 0
Root: x = 4 / 2 = 2 (repeated)
Factored form: (x - 2)² = 0
Equation: x² + 2x + 5 = 0 (a=1, b=2, c=5)
Discriminant: (2)² - 4(1)(5) = 4 - 20 = -16 < 0
Roots: x = [-2 ± 4i] / 2 = -1 ± 2i
Complex conjugate pair
Applications in Real Life
- Physics: Projectile motion, free fall, collision problems
- Engineering: Bridge design, structural analysis, circuit design
- Economics: Profit maximization, cost analysis, break-even analysis
- Biology: Population modeling, growth rates
- Geometry: Area and perimeter problems, optimization
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Frequently Asked Questions
1. What if coefficient a is zero?
If a = 0, it's not a quadratic equation—it becomes linear (bx + c = 0). The quadratic formula doesn't apply.
2. What are complex roots?
Complex roots occur when the discriminant is negative. They involve imaginary units (i) and come in conjugate pairs: a ± bi.
3. How do you find the vertex of a parabola?
The vertex's x-coordinate is -b/(2a). Substitute this into the equation to find the y-coordinate. The vertex is the minimum or maximum point.
4. What is the significance of the discriminant?
The discriminant (b² - 4ac) determines the number and type of roots without calculating them. It tells you the solution's nature before solving.
5. Can all quadratic equations be factored?
No. Only equations with rational roots factor nicely. Equations with irrational or complex roots require the quadratic formula.
6. What is completing the square?
A technique to convert ax² + bx + c into a(x - h)² + k form. Useful for finding the vertex and understanding parabola transformations.
7. How do Vieta's formulas help?
Vieta's formulas relate coefficients to root sums and products: sum = -b/a, product = c/a. Useful for finding equations given roots.
8. What does the parabola direction depend on?
The parabola opens upward (U-shape) if a > 0 and downward (inverted U) if a < 0. The magnitude of a affects how "wide" or "narrow" it is.
9. Can a quadratic equation have no real solutions?
Yes. When the discriminant is negative, there are no real solutions—only complex solutions. Graphically, the parabola doesn't cross the x-axis.
10. How do you verify your solutions?
Substitute each root back into the original equation. If both sides equal zero, the solution is correct.
11. What is the standard form vs vertex form?
Standard form: ax² + bx + c. Vertex form: a(x - h)² + k. Vertex form clearly shows the vertex (h, k) and makes graphing easier.
12. Is this solver accurate for all equations?
Yes, the quadratic formula is a general method that works for all quadratic equations with real or complex coefficients.