Welcome to our specialized Quadratic Equation Solver, designed to mimic the precision and functionality you'd expect from a Texas Instruments scientific calculator. Quadratic equations are fundamental in mathematics, physics, engineering, and economics, describing parabolas and various real-world phenomena.
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 squared but no term with a higher power. The standard form of a quadratic equation is:
ax² + bx + c = 0
Where:
a, b, and c are coefficients, with a ≠ 0.
x represents the unknown variable.
The solutions for x are also known as the roots of the equation, or the points where the parabola intersects the x-axis.
The Quadratic Formula
The roots of a quadratic equation can be found using the quadratic formula:
x = [-b ± √(b² - 4ac)] / 2a
The term (b² - 4ac) is called the discriminant (Δ), and its value determines the nature of the roots:
If Δ > 0: There are two distinct real roots.
If Δ = 0: There is exactly one real root (a repeated root).
If Δ < 0: There are two complex conjugate roots.
How to Use This Calculator
Our Quadratic Equation Solver simplifies the process of finding these roots. Simply input the coefficients a, b, and c from your quadratic equation into the respective fields:
Coefficient a: Enter the number multiplying the x² term. (e.g., for 2x² + 3x - 5 = 0, enter 2)
Coefficient b: Enter the number multiplying the x term. (e.g., for 2x² + 3x - 5 = 0, enter 3)
Coefficient c: Enter the constant term. (e.g., for 2x² + 3x - 5 = 0, enter -5)
Click the "Calculate Roots" button, and the calculator will instantly display the solutions for x, whether they are real or complex.
Examples
Example 1: Two Distinct Real Roots
Consider the equation: x² - 3x + 2 = 0
Coefficient a: 1
Coefficient b: -3
Coefficient c: 2
Using the calculator, you would input these values. The discriminant would be (-3)² - 4(1)(2) = 9 - 8 = 1. Since Δ > 0, there are two real roots:
x1 = [3 + √1] / 2 = (3 + 1) / 2 = 2
x2 = [3 - √1] / 2 = (3 - 1) / 2 = 1
The calculator will output: x1 = 2.0000, x2 = 1.0000
Example 2: One Real Root (Repeated)
Consider the equation: x² - 4x + 4 = 0
Coefficient a: 1
Coefficient b: -4
Coefficient c: 4
The discriminant would be (-4)² - 4(1)(4) = 16 - 16 = 0. Since Δ = 0, there is one real root:
x = [4 ± √0] / 2 = 4 / 2 = 2
The calculator will output: x = 2.0000
Example 3: Two Complex Conjugate Roots
Consider the equation: x² + 2x + 5 = 0
Coefficient a: 1
Coefficient b: 2
Coefficient c: 5
The discriminant would be (2)² - 4(1)(5) = 4 - 20 = -16. Since Δ < 0, there are two complex conjugate roots:
Real Part = -2 / (2*1) = -1
Imaginary Part = √|-16| / (2*1) = 4 / 2 = 2
The calculator will output: x1 = -1.0000 + 2.0000i, x2 = -1.0000 – 2.0000i
This tool is perfect for students, educators, and professionals who need quick and accurate solutions to quadratic equations, mirroring the reliability of a dedicated scientific calculator.