Interactive Quadratic Equation Solver
Solve equations of the form ax² + bx + c = 0 with step-by-step solutions
Solution Methods
Learn three powerful methods to solve quadratic equations
1. Quadratic Formula
The most general method that works for any quadratic equation. Always produces a solution when one exists.
x = (-b ± √(b² - 4ac)) / (2a)
2. Factoring
The fastest method when the quadratic can be factored into two linear expressions. Works well with integer solutions.
ax² + bx + c = (px + q)(rx + s)
3. Completing the Square
Transform the equation into a perfect square trinomial. Useful for understanding the structure of quadratics.
a(x - h)² + k = 0
Worked Examples
Method 1: Quadratic Formula
2x² - 7x + 3 = 0
1
Identify coefficients: a = 2, b = -7, c = 3
2
Calculate discriminant: Δ = b² - 4ac = (-7)² - 4(2)(3) = 49 - 24 = 25
3
Apply quadratic formula: x = (7 ± √25) / (2·2) = (7 ± 5) / 4
4
Calculate solutions: x₁ = (7 + 5) / 4 = 3 and x₂ = (7 - 5) / 4 = 0.5
Answer: x = 3 or x = 0.5
Key Concepts
Discriminant
The expression b² - 4ac determines the nature of solutions:
- Positive: Two real solutions
- Zero: One repeated solution
- Negative: Two complex solutions
Vertex Form
Standard form ax² + bx + c can be written as a(x - h)² + k where (h, k) is the vertex.
h = -b/(2a), k = f(h)
Roots and Factors
If r₁ and r₂ are roots, then the quadratic can be written as:
a(x - r₁)(x - r₂) = 0