ALGEBRA NOTES:

Quadratic formula:

Often, the simplest way to solve "ax² + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. But sometimes the quadratic is too messy, or it doesn't factor at all, or you just don't feel like factoring. So, while factoring may not always be successful, the Quadratic Formula can always find the solution.

The Quadratic Formula uses the "a", "b", and "c" from "ax² + bx + c", where "a", "b", and "c" are just numbers. The Formula is derived from the process of completing the square, and is formally stated as: For ax² + bx + c = 0, the value of x is given by

Note that, for the Formula to work, you must have "(quadratic) = 0". Note also that the "2a" at the bottom of the Formula is underneath everything above, not just the square root. And don't forget that it's a "2a" under there, not just a "2"! And make sure that you are careful not to drop the square root or the "plus/minus" in the middle of your calculations, or I can guarantee that you will forget to "put them back" on your test, and you'll mess yourself up. And remember that "b²" means "the square of ALL of b, including the sign", so don't leave b² being negative, even if b is negative, because the square of a negative is a positive.

In other words, don't be sloppy and don't try to take shortcuts, because it will only hurt you in the long run. Trust me on this!

 

The Discriminant:

The expression b²-4ac in the quadratic equation is called the discriminant. The discriminant gives you information about the number of possible solutions when you solve a quadratic equation.

The disceriminant tells you how many real solutions there are of an equation in the form ax² + bx + c = 0 when a < > 0.

When b² - 4ac = 0, there is exactly one solution. The graph of y=ax²+bx+c has one x-intercept.

When b² - 4ac > 0, there are two solutions. The graph of y=ax²+bx+c has two x-intercepts.

When b² - 4ac < 0, there are no solutions. The graph of y=ax²+bx+c has no x-intercepts.