Sooner or later, you’ll try to solve a quadratic equation that looks something like
this: 5x^2 – 3x – 4 = 0. Sometime after the thirtieth try, you’ll realize that it
doesn’t factor nicely. That’s where the quadratic formula comes in.
With a trinomial in the form ax^2 + bx + c = 0, the two roots are:
(If you need to convince yourself, try this using a trinomial that you know
factors nicely.)
Now back to the more difficult equation:
Solve for x if 5x^2 – 3x – 4 = 0In this trinomial, a = 5, b = –3, and c = –4.
Plug these into the formula.So the two factors areThe great thing about the quadratic formula is that it will always give you
some kind of solution for x, whereas the binomial factoring method doesn’t
always work out (sometimes the product of a × c doesn’t factor into two
numbers that add up to b). Although the quadratic formula is harder to
remember, if you practice it enough times, it’s the safer choice. Just be extra
careful with your signs when evaluating the b^2 – 4ac part (the part under the
square root). Our private research indicates that this is the source of 87.42% of
all quadratic formula fails.
Functions
A function is just like any other algebraic expression, except we record the input
and the output, so that we can graph it on a coordinate plane.