Quadratic Formula Solver Formula
Understand the math behind the quadratic formula solver. Each variable explained with a worked example.
Formulas Used
Discriminant
discriminant = pow(b, 2) - 4 * a * cRoot1
root1 = a != 0 ? (-b + sqrt(abs(pow(b, 2) - 4 * a * c))) / (2 * a) : 0Root2
root2 = a != 0 ? (-b - sqrt(abs(pow(b, 2) - 4 * a * c))) / (2 * a) : 0Vertex X
vertex_x = a != 0 ? -b / (2 * a) : 0Vertex Y
vertex_y = a != 0 ? c - pow(b, 2) / (4 * a) : 0Variables
| Variable | Description | Default |
|---|---|---|
a | Coefficient a | 1 |
b | Coefficient b | -5 |
c | Coefficient c | 6 |
How It Works
How to Solve a Quadratic Equation
Quadratic Formula
x = (-b ± sqrt(b² - 4ac)) / (2a)
For the equation ax² + bx + c = 0:
1. Calculate the discriminant: D = b² - 4ac 2. If D > 0: two distinct real roots 3. If D = 0: one repeated real root 4. If D < 0: no real roots (complex roots)
Vertex
The vertex of the parabola is at x = -b/(2a), y = c - b²/(4a).
Worked Example
Solve x² - 5x + 6 = 0.
- 01D = (-5)² - 4(1)(6) = 25 - 24 = 1
- 02x₁ = (5 + √1) / 2 = 6/2 = 3
- 03x₂ = (5 - √1) / 2 = 4/2 = 2
- 04Roots are x = 3 and x = 2
Frequently Asked Questions
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / (2a). It gives the solutions to any quadratic equation ax² + bx + c = 0.
What does the discriminant tell you?
The discriminant (b² - 4ac) reveals the nature of the roots: positive means two real roots, zero means one repeated root, negative means complex roots.
What if a = 0?
If a = 0, the equation is linear (bx + c = 0), not quadratic. The solution is x = -c/b.
Learn More
Guide
How to Solve Quadratic Equations - Complete Guide
Learn how to solve quadratic equations using factoring, the quadratic formula, and completing the square. Step-by-step methods with worked examples.
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