Enter Coefficients — ax² + bx + c = 0
Enter the values of a, b, and c (a cannot be zero)
ax² + bx + c = 0
Cannot be 0
x² +
Can be 0
x +
Can be 0
= 0
Examples:
Quadratic Formula & How to Solve Quadratic Equations
A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. Every quadratic has exactly two solutions (roots), which may be real or complex numbers.
x = (−b ± √(b²−4ac)) / (2a)
Discriminant: Δ = b² − 4ac
Δ > 0 → Two real roots | Δ = 0 → One repeated root | Δ < 0 → Two complex roots
Discriminant: Δ = b² − 4ac
Δ > 0 → Two real roots | Δ = 0 → One repeated root | Δ < 0 → Two complex roots
Three Methods to Solve Quadratic Equations
- Quadratic Formula: Always works for any quadratic. Use when other methods are difficult.
- Factoring: Fastest when the discriminant is a perfect square (integer roots or simple fractions). Example: x²−5x+6=(x−2)(x−3)=0
- Completing the Square: Most useful for deriving the formula or when a=1. Converts to (x+p)²=q form.
Vertex and Parabola Properties
Vertex: h = −b/(2a) | k = f(h) = ah² + bh + c
Axis of Symmetry: x = h
Opens Up if a > 0 (minimum) | Opens Down if a < 0 (maximum)
Axis of Symmetry: x = h
Opens Up if a > 0 (minimum) | Opens Down if a < 0 (maximum)
Sum and Product of Roots (Vieta's Formulas)
Sum of roots: x₁ + x₂ = −b/a
Product of roots: x₁ × x₂ = c/a
Product of roots: x₁ × x₂ = c/a
Real-World Uses of Quadratic Equations
- Projectile motion: Height of a ball thrown in the air (h = −16t² + vt + h₀)
- Area problems: Finding dimensions of rectangles with given area
- Profit/revenue: Business models with quadratic cost functions
- Physics: Acceleration, kinematic equations
- Engineering: Parabolic bridges, satellite dishes, reflectors
Quadratic Equation Examples
x² − 5x + 6 = 0 → x = 3, x = 2 (Δ = 1, two rational roots)
x² − 6x + 9 = 0 → x = 3 (Δ = 0, one repeated root)
x² + 2x + 5 = 0 → x = −1 ± 2i (Δ = −16, complex roots)
2x² + 5x − 3 = 0 → x = ½, x = −3 (Δ = 49, two rational roots)
x² − 6x + 9 = 0 → x = 3 (Δ = 0, one repeated root)
x² + 2x + 5 = 0 → x = −1 ± 2i (Δ = −16, complex roots)
2x² + 5x − 3 = 0 → x = ½, x = −3 (Δ = 49, two rational roots)
Completing the Square — Step by Step
Completing the square rewrites ax²+bx+c=0 into vertex form:
x² + bx + c = 0
→ x² + bx = −c
→ x² + bx + (b/2)² = −c + (b/2)²
→ (x + b/2)² = (b²−4c)/4
→ x = −b/2 ± √(b²−4c)/2
→ x² + bx = −c
→ x² + bx + (b/2)² = −c + (b/2)²
→ (x + b/2)² = (b²−4c)/4
→ x = −b/2 ± √(b²−4c)/2
Nature of Roots — Quick Reference
- Δ > 0 and perfect square: Two rational roots — can be factored
- Δ > 0 and not perfect square: Two irrational roots — use quadratic formula
- Δ = 0: One repeated rational root — parabola touches x-axis once
- Δ < 0: Two complex (imaginary) roots — parabola doesn't cross x-axis
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