Quadratic formula
Often, the simplest way to solve "ax^2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to 0, 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 might not usually be successful, the Quadratic Formula could usually find the solution.
The Quadratic Formula uses the "a", "b", and "c" from "ax^2 + 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 could guarantee that you will forget to "put them back" on your test, and you will 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 negati
ve 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!
Solve x2 + 3x – 4 = 0
Note first that this quadratic happens to factor:
x2 + 3x – 4 = (x + 4)(x – 1) = 0
...so x = –4 and x = 1. How would this look in the Quadratic Formula? Using a = 1, b = 3, and c = –4, it looks like this:
Recall that, when y = 0, you are finding the x-intercepts of the graph. So solving ax2 + bx + c = 0 for x means that, among other things, you are trying to find the x-intercepts. Since you came up with two solutions for this equation, there must be two x-intercepts on the graph. Graphing, you get the curve at the left:
As you could see, the x-intercepts match the solutions, falling at x = –4 and x = 1. This shows the connection between graphing and solving: When you are solving "(quadratic) = 0", you are finding the x-intercepts of the graph. This could be useful if you have a graphing calculator, because you can use the Quadratic Formula (when necessary) to solve a quadratic, and then use your graphing calculator to make sure that the displayed x-intercepts have the same decimal values as the solutions that the Quadratic Formula gives you. (Note that the calculator display on the graph will probably have some pixel-related round-off error, so you'd be checking to see that the values were close; don't expect them to be exact on the screen.)
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