Quadratic Equation Solver
Enter the coefficients a, b, and c for the quadratic equation in the standard form ax² + bx + c = 0.
Understanding Quadratic Equations
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:
- a, b, and c are coefficients (constants), with 'a' not equal to zero.
- x represents the unknown variable.
The solutions to a quadratic equation are called its roots. These roots represent the x-intercepts of the parabola that the quadratic equation forms when graphed.
The Quadratic Formula
The most common method to find the roots of a quadratic equation is by using the quadratic formula:
x = [-b ± sqrt(b² – 4ac)] / (2a)
This formula allows you to directly calculate the values of 'x' once you know the coefficients a, b, and c.
The Discriminant (b² – 4ac)
A crucial part of the quadratic formula is the expression under the square root, b² – 4ac, which is known as the discriminant (often denoted by the Greek letter Delta, Δ). The value of the discriminant tells us about the nature of the roots:
- If Δ > 0 (positive): There are two distinct real roots. This means the parabola intersects the x-axis at two different points.
- If Δ = 0: There is exactly one real root (also called a repeated or double root). The parabola touches the x-axis at exactly one point.
- If Δ < 0 (negative): There are two complex (or imaginary) roots. The parabola does not intersect the x-axis. Complex roots always come in conjugate pairs (a + bi, a – bi).
How to Use the Calculator
Our Quadratic Equation Solver simplifies the process of finding roots. Simply input the coefficients 'a', 'b', and 'c' from your equation (e.g., for 2x² + 5x – 3 = 0, you would enter a=2, b=5, c=-3) and click "Calculate Roots". The calculator will instantly provide the real or complex solutions.
Examples
Let's look at some examples to illustrate the different types of roots:
Example 1: Two Distinct Real Roots
Equation: x² – 3x + 2 = 0
- a = 1
- b = -3
- c = 2
Discriminant = (-3)² – 4(1)(2) = 9 – 8 = 1 (positive)
Roots: x1 = [-(-3) + sqrt(1)] / (2*1) = (3 + 1) / 2 = 2
Roots: x2 = [-(-3) – sqrt(1)] / (2*1) = (3 – 1) / 2 = 1
Calculator Output: x1 = 2.0000, x2 = 1.0000
Example 2: One Real (Repeated) Root
Equation: x² – 4x + 4 = 0
- a = 1
- b = -4
- c = 4
Discriminant = (-4)² – 4(1)(4) = 16 – 16 = 0
Root: x = [-(-4)] / (2*1) = 4 / 2 = 2
Calculator Output: x = 2.0000
Example 3: Two Complex Roots
Equation: x² + 2x + 5 = 0
- a = 1
- b = 2
- c = 5
Discriminant = (2)² – 4(1)(5) = 4 – 20 = -16 (negative)
Roots: x1 = [-2 + sqrt(-16)] / (2*1) = (-2 + 4i) / 2 = -1 + 2i
Roots: x2 = [-2 – sqrt(-16)] / (2*1) = (-2 – 4i) / 2 = -1 – 2i
Calculator Output: x1 = -1.0000 + 2.0000i, x2 = -1.0000 – 2.0000i
Example 4: Linear Equation (a=0)
Equation: 0x² + 5x – 10 = 0 (or simply 5x – 10 = 0)
- a = 0
- b = 5
- c = -10
Root: 5x = 10 => x = 2
Calculator Output: This is a linear equation (a=0). The root is: x = 2.0000