A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. It is called "quadratic" because the variable is raised to the second power.
The quadratic formula is a formula used to solve quadratic equations. It is given by x = (-b ± √(b^2 - 4ac)) / 2a. It is useful when the equation does not factor easily.
There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. To solve by factoring, find two numbers that when multiplied together equal c and when added together equal b. Then use those numbers to rewrite the equation as (ax + m)(ax + n) = 0 and solve for x. To solve by completing the square, follow a specific set of steps to rewrite the equation in the form (x + p)^2 = q and solve for x. To solve using the quadratic formula, simply plug in the values for a, b, and c and solve for x.
A quadratic equation can have two real solutions, one real solution, or two complex solutions. Two real solutions occur when the discriminant (b^2 - 4ac) is positive, one real solution occurs when the discriminant is 0, and two complex solutions occur when the discriminant is negative.
A quadratic equation has solutions if the discriminant (b^2 - 4ac) is greater than or equal to 0. If the discriminant is 0, the equation has one real solution. If the discriminant is positive, the equation has two real solutions. If the discriminant is negative, the equation has two complex solutions.