A quadratic equation solver finds the solutions to a quadratic equation of the form y = ax² + bx + c, where a is not equal to 0. It finds the values of x in this equation to make y = 0 when plugging in those values of x.
Quadratic equation solvers commonly determine the values of x that results in y = 0 by using the quadratic formula, x = (-b + or - sqrt(b² - 4ac)) / (2a). The portion of the quadratic formula, b² - 4ac, is known as the discriminant that determines the number of real values for x that solves the equation 0 = ax² + bx + c. If b² - 4ac is greater than 0, then there are two real solutions for x. If b² - 4ac = 0, then there is one real solution for x. If b² - 4ac < 0, then there is no real solution for x, because sqrt(b² - 4ac) is an imaginary number.
Other quadratic equation solvers determine the values for x by completing the square rather than using the quadratic formula. They manipulate the equation so that one side of the equation is a quadratic expression that is a perfect square that can be factored, and the rest of the terms are moved to the other side of the equation. For example, in the quadratic equation x² + 2x + 3 = 0, the solver subtracts 3 from both sides, then adds 1 to both sides to form x² + 2x + 1 = -2, or (x+1)² = -2. Then x + 1 = + or - sqrt(-2), so x = -1 + or - i*sqrt(2), where "i" is the imaginary square root of -1.