Quadratic Formula Calculator — Step-by-Step Solutions
Enter coefficients for ax² + bx + c = 0
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The Quadratic Formula Calculator solves equations of the form ax² + bx + c = 0 using the quadratic formula. It finds both real and complex roots, calculates the discriminant, and identifies the vertex of the parabola.
Enter the coefficients a, b, and c to get instant solutions. The calculator shows the discriminant value to determine whether roots are real, repeated, or complex.
How to Use the Quadratic Formula Calculator
- Enter the coefficients a, b, and c from the equation ax^2 + bx + c = 0.
- The coefficient a must not be zero (otherwise it is not a quadratic equation).
- The calculator finds the roots, discriminant, vertex, and axis of symmetry.
Formula and Theory
The quadratic formula solves any equation of the form ax^2 + bx + c = 0. The discriminant (b^2 - 4ac) determines the nature of the roots: positive means two distinct real roots, zero means one repeated root, negative means two complex conjugate roots.
The vertex of the parabola y = ax^2 + bx + c is at (-b/2a, f(-b/2a)). The axis of symmetry is the vertical line x = -b/2a.
Worked Examples
Example 1: Two real roots
Problem: Solve x^2 - 5x + 6 = 0
Solution: a=1, b=-5, c=6. Discriminant = 25-24 = 1. x = (5 +/- 1)/2
Answer: x = 3 or x = 2
Example 2: One repeated root
Problem: Solve x^2 - 6x + 9 = 0
Solution: Discriminant = 36 - 36 = 0. x = 6/2
Answer: x = 3 (repeated)
Example 3: Complex roots
Problem: Solve x^2 + 4 = 0
Solution: a=1, b=0, c=4. Discriminant = 0 - 16 = -16. x = +/- sqrt(-16)/2
Answer: x = 2i or x = -2i
Frequently Asked Questions
What is the discriminant?
Can the quadratic formula solve any quadratic equation?
What other methods can solve quadratic equations?
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