Quadratic Formula Calculator
Solves equations in the form ax² + bx + c = 0
The solution(s) for x are:
Discriminant (b² - 4ac):
How Does This Calculator Work?
This calculator solves a quadratic equation by applying the well-known quadratic formula. A quadratic equation has the general form ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients.
x = [ -b ± √(b² - 4ac) ] / 2a
- The Discriminant: The part of the formula under the square root, b² - 4ac, is called the discriminant. Its value determines the nature of the solutions (or roots).
- Two Real Roots: If the discriminant is positive, there are two distinct real solutions.
- One Real Root: If the discriminant is zero, there is exactly one real solution.
- Two Complex Roots: If the discriminant is negative, there are no real solutions, but there are two complex solutions involving imaginary numbers. This calculator will show these complex roots.
The Surprising History of Solving Quadratics
Solving quadratic equations is a cornerstone of algebra, but its history stretches back long before the quadratic formula was written in its modern form. Ancient Babylonian mathematicians, as early as 2000 BC, were able to solve quadratic problems using a method of "completing the square." They didn't have algebraic notation, so they described their methods through step-by-step word problems related to land area and surveying.
Greek mathematicians like Euclid used geometric methods to solve these problems. However, it was the 9th-century Persian mathematician, Muhammad ibn Musa al-Khwarizmi (from whose name we get the word "algorithm"), who provided a comprehensive and systematic approach to solving linear and quadratic equations in his influential book, *The Compendious Book on Calculation by Completion and Balancing*. He provided general rules for solving them, though still expressed in words rather than symbols. The modern symbolic formula we use today wasn't fully developed until the 16th and 17th centuries.
Explore More Related Tools
While you're here, check out some of our other popular math and algebra calculators:
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Frequently Asked Questions (FAQ)
What are the 'roots' of an equation?
The roots, or solutions, of a quadratic equation are the values of 'x' that make the equation true (i.e., make the expression equal to zero). Geometrically, these are the points where the parabola representing the equation crosses the x-axis.
What are complex roots?
When the discriminant (b² - 4ac) is negative, the formula requires taking the square root of a negative number. In the real number system, this is impossible. The solutions are therefore "complex numbers," which include the imaginary unit 'i' (where i = √-1). This calculator will display these complex roots.
What if my equation doesn't look like ax² + bx + c = 0?
You must first rearrange your equation into this standard form. For example, if you have x² + 3x = 4, you must first subtract 4 from both sides to get x² + 3x - 4 = 0. Now you can see that a=1, b=3, and c=-4.