Math Calculator with Solution

Quadratic Equation Solver

Enter the coefficients a, b, and c for your quadratic equation in the standard form ax² + bx + c = 0 to find its roots and a step-by-step solution.

Coefficient 'a': Coefficient 'b': Coefficient 'c':

Understanding the Quadratic Equation and Its Solutions

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the power of two. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where:

x represents the unknown variable.
a, b, and c are coefficients, with a not equal to zero. If a were zero, the equation would become a linear equation (bx + c = 0).

The Quadratic Formula

The most common method to find the roots (solutions) of a quadratic equation is by using the quadratic formula. The roots are the values of x that satisfy the equation. The formula is:

x = [-b ± sqrt(b² - 4ac)] / (2a)

This formula provides two potential solutions due to the "±" (plus or minus) sign, which accounts for the two possible square roots of a number.

The Discriminant (Δ)

A crucial part of the quadratic formula is the expression under the square root sign: b² - 4ac. This is called the discriminant, often denoted by the Greek letter Delta (Δ). The value of the discriminant tells us about the nature of the roots without actually calculating them:

If Δ > 0 (Discriminant is positive): The equation has two distinct real roots. This means there are two different numerical solutions for x. Graphically, the parabola intersects the x-axis at two different points.
If Δ = 0 (Discriminant is zero): The equation has exactly one real root (also called a repeated or double root). This means both solutions from the quadratic formula are identical. Graphically, the parabola touches the x-axis at exactly one point (its vertex).
If Δ < 0 (Discriminant is negative): The equation has two complex conjugate roots. This means there are no real number solutions for x. Instead, the solutions involve imaginary numbers. Graphically, the parabola does not intersect the x-axis at all.

How to Use the Calculator

Our Quadratic Equation Solver simplifies the process of finding roots and understanding the solution steps:

Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember to include their signs (e.g., if you have x² - 3x + 2 = 0, then a=1, b=-3, c=2).
Enter Values: Input these values into the respective fields in the calculator.
Calculate: Click the "Calculate Roots" button.
Review Results: The calculator will display the roots (x₁ and x₂) and a detailed step-by-step breakdown of how those roots were derived, including the calculation of the discriminant and its interpretation.

Examples

Example 1: Two Distinct Real Roots

Consider the equation: x² - 5x + 6 = 0

a = 1
b = -5
c = 6

Discriminant (Δ): (-5)² - 4 * 1 * 6 = 25 - 24 = 1

Since Δ > 0, there are two distinct real roots.

Roots:

x₁ = [5 + sqrt(1)] / (2 * 1) = (5 + 1) / 2 = 6 / 2 = 3
x₂ = [5 - sqrt(1)] / (2 * 1) = (5 - 1) / 2 = 4 / 2 = 2

The roots are 3 and 2.

Example 2: One Real (Repeated) Root

Consider the equation: x² - 4x + 4 = 0

a = 1
b = -4
c = 4

Discriminant (Δ): (-4)² - 4 * 1 * 4 = 16 - 16 = 0

Since Δ = 0, there is one real (repeated) root.

Root:

x = [4 ± sqrt(0)] / (2 * 1) = 4 / 2 = 2

The root is 2.

Example 3: Two Complex Conjugate Roots

Consider the equation: x² + 2x + 5 = 0

a = 1
b = 2
c = 5

Discriminant (Δ): (2)² - 4 * 1 * 5 = 4 - 20 = -16

Since Δ < 0, there are two complex conjugate roots.