Quadratic Formula Calculator
Solve quadratic equations of the form ax² + bx + c = 0 using the quadratic formula. Get step-by-step solutions, graph visualization, and downloadable results.
Input Values
Solution
Equation: x² - 3x + 2 = 0
Discriminant (D) = b² - 4ac = 1
Roots: x = 2, x = 1
Solution Steps
Using quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)
Tip
When the discriminant is positive, you get two real roots.
Frequently Asked Questions
The quadratic formula is used to solve quadratic equations of the form ax² + bx + c = 0. The formula is x = (-b ± √(b² - 4ac)) / (2a).
The discriminant (b² - 4ac) determines the nature of the roots:
- If D > 0: Two distinct real roots
- If D = 0: One real root (repeated)
- If D < 0: Two complex roots
Yes, this calculator will display complex roots in the form a ± bi when the discriminant is negative.
If a = 0, the equation is not quadratic but linear. The calculator will show an error message in this case.
Understanding Quadratic Equations and the Quadratic Formula
Quadratic equations are polynomial equations of the second degree, meaning they contain at least one term that is squared. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are coefficients and x represents the variable.
These equations are fundamental in algebra and have wide-ranging applications in physics, engineering, economics, and many other fields. They can describe the trajectory of a projectile, model profit functions in business, or define the shape of a satellite dish.
The solutions to quadratic equations are called roots, and they represent the x-values where the parabola (the graph of a quadratic equation) intersects the x-axis. A quadratic equation can have either two real roots, one real root, or two complex roots, depending on the value of the discriminant (b² - 4ac).
The quadratic formula provides a straightforward method to find these roots, regardless of whether they are real or complex. This formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0.
While factoring is another method to solve quadratics, it only works easily when the roots are rational numbers. The quadratic formula, on the other hand, works for any quadratic equation with real or complex coefficients.
Understanding how to use and apply the quadratic formula is an essential skill in mathematics. This calculator simplifies the process, providing not only the solutions but also showing the steps involved and visualizing the graph of the equation.
When using the quadratic formula, it's important to pay attention to the sign of the discriminant. A positive discriminant indicates two distinct real roots, a zero discriminant means exactly one real root (a repeated root), and a negative discriminant results in two complex conjugate roots.
This calculator also provides a graph of the quadratic function, which helps in visualizing the solutions. The roots are the points where the parabola crosses the x-axis. When there are no real roots (negative discriminant), the parabola does not intersect the x-axis at all.
Whether you're a student learning algebra, a professional needing quick calculations, or someone refreshing their math skills, this quadratic formula calculator provides a convenient tool to solve quadratic equations accurately and efficiently.