Quadratic Formula Calculator
Solve quadratic equations of the form ax² + bx + c = 0 instantly.
Understanding the Quadratic Formula
A quadratic equation is a second-order polynomial equation in a single variable x, expressed in the standard form:
ax² + bx + c = 0
The Quadratic Formula is the most reliable method to find the roots (solutions) of such equations, regardless of whether they can be factored easily. The formula is:
x = [-b ± √(b² – 4ac)] / 2a
Components of the Formula
- a: The quadratic coefficient (the number in front of x²). It cannot be zero.
- b: The linear coefficient (the number in front of x).
- c: The constant term.
- Discriminant (Δ): The part inside the square root (b² – 4ac). This value determines the nature of the roots.
Types of Solutions
Based on the value of the discriminant, you will encounter three scenarios:
| Discriminant Value | Nature of Roots |
|---|---|
| Positive (Δ > 0) | Two distinct real numbers. |
| Zero (Δ = 0) | One repeated real number. |
| Negative (Δ < 0) | Two complex (imaginary) numbers. |
Step-by-Step Example
Let's solve the equation: x² – 5x + 6 = 0
- Identify Coefficients: Here, a = 1, b = -5, and c = 6.
- Calculate Discriminant: Δ = (-5)² – 4(1)(6) = 25 – 24 = 1.
- Apply Formula: Since Δ is positive, we use x = [-(-5) ± √1] / 2(1).
- Solve: x = (5 ± 1) / 2. This gives us x = 6/2 = 3 and x = 4/2 = 2.
- Result: The solutions are x = 2 and x = 3.
Frequently Asked Questions
What happens if 'a' is zero?
If a = 0, the equation is no longer quadratic; it becomes a linear equation (bx + c = 0). You can solve it by isolating x: x = -c/b.
Can the quadratic formula solve all quadratic equations?
Yes, it is a universal method. While factoring or completing the square might be faster for some equations, the quadratic formula works for every single quadratic equation, including those with irrational or complex roots.