Quadratic Formula Calculator - Solve Equations with Steps

📐 Quadratic Formula Calculator

Solve quadratic equations of the form ax² + bx + c = 0 instantly

Standard Equation: ax² + bx + c = 0

Coefficient (a):

Coefficient (b):

Constant (c):

Discriminant (Δ = b² - 4ac): -

Nature of Roots: -

---

First Root (x₁):

-

Second Root (x₂):

-

Understanding the Quadratic Formula in Algebra

A quadratic equation is a second-order polynomial equation in a single variable x, with the standard form ax² + bx + c = 0, where a ≠ 0. The quadratic formula provides the algebraic solution to find the roots (or zeros) of any quadratic equation:

x = [ -b ± √(b² - 4ac) ] / 2a

The Role of the Discriminant (b² - 4ac)

The term inside the square root, Δ = b² - 4ac, is known as the discriminant. It determines the number and real or complex nature of the roots:

Discriminant Value	Nature of Roots	Graph Result (Parabola)
Δ > 0 (Positive)	Two distinct real roots	Crosses the X-axis at two distinct points.
Δ = 0 (Zero)	One real repeated root	Touches the X-axis at exactly one vertex point.
Δ < 0 (Negative)	Two complex / imaginary roots (i)	Does not intersect the X-axis.

Step-by-Step Example Calculation

Let's solve the quadratic equation x² - 5x + 6 = 0:

1. Identify coefficients: a = 1, b = -5, c = 6.
2. Calculate Discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1.
3. Since Δ > 0, there are two real roots.
4. Calculate Roots: x = [ 5 ± √1 ] / 2 = [ 5 ± 1 ] / 2.
5. x₁ = 6 / 2 = 3 and x₂ = 4 / 2 = 2.

Frequently Asked Questions (FAQs)

Q: What happens if 'a' equals zero?
A: If a = 0, the equation is no longer quadratic; it becomes a linear equation (bx + c = 0).

Q: What is an imaginary root?
A: When the discriminant is negative, taking the square root of a negative number introduces the imaginary unit i (where i = √-1).