Math & Engineering
Quadratic Formula Calculator
Calculate the roots of any quadratic equation using the quadratic formula. Enter the coefficients a, b, and c to find both real and complex solutions.
Enter coefficients and click Calculate to see results
Related to Quadratic Formula Calculator
The quadratic formula calculator solves quadratic equations in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The calculator uses the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a) to find the roots (solutions) of the equation.
Understanding the Discriminant
The discriminant (b² - 4ac) determines the nature of the roots:
- If discriminant > 0: Two distinct real roots
- If discriminant = 0: One repeated real root
- If discriminant < 0: Two complex conjugate roots
Complex Numbers
When the discriminant is negative, the calculator expresses the roots in the form a ± bi, where a is the real part and bi is the imaginary part. The letter i represents the imaginary unit, defined as i² = -1.
The calculator provides comprehensive results for any quadratic equation. Here's how to interpret them:
Real Roots
When the roots are real numbers, they represent the x-values where the parabola crosses the x-axis. These points are also called zeros or x-intercepts of the quadratic function.
Complex Roots
Complex roots occur in conjugate pairs and indicate that the parabola never crosses the x-axis. The real part represents the x-coordinate of the axis of symmetry, while the imaginary part indicates how far the parabola "misses" the x-axis.
1. What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2, written in the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The term ax² is called the quadratic term, bx is the linear term, and c is the constant term.
2. Why can't the coefficient 'a' be zero?
If a = 0, the equation becomes bx + c = 0, which is a linear equation, not a quadratic equation. The quadratic formula only works when a ≠ 0, as division by zero is undefined.
3. What does the discriminant tell us?
The discriminant (b² - 4ac) reveals the nature of the roots without solving the equation. A positive discriminant means two real roots, zero means one repeated real root, and a negative discriminant indicates complex conjugate roots.
4. How accurate are the calculated roots?
The calculator provides results rounded to 4 decimal places for clarity. The actual computation is performed with high precision using JavaScript's native Math functions, ensuring accurate results for most practical applications.
5. What is the scientific source for this calculator?
The quadratic formula calculator is based on the fundamental algebraic solution derived from completing the square method, first documented in ancient Babylonian mathematics (circa 2000 BCE). The modern form of the quadratic formula was established in the standard mathematical literature through works such as Descartes' "La Géométrie" (1637) and Euler's "Elements of Algebra" (1765). The implementation follows the rigorous mathematical principles outlined in contemporary algebra textbooks and adheres to IEEE 754 floating-point arithmetic standards for numerical computations.