We've done a variety of factoring problems by this point.
Now factoring is great but the problem with factoring is that it doesn't always work.
There are somethings that just don't factor no matter what you do.
We actually have a formula that takes care of that for us, the quadratic formula.
So what you do is you look at these coefficients ab and c and the quadratic formula is made from those.
If you just have negative b plus or minus the square root of b squared minus 4ac all over 2a.
Now you will need to memorize this formula, it's not quiet as hard as it looks though, it just takes a little
practice, we'll even start right now with these two examples.
So for example one we have 6x squared plus 7x minus 3 equals 0, in this case a equals 6, b equals 7 and c equals
negative 3 and we'll just plug them on in.
Now remember that you do need to use parenthesis when you substitute numbers in.
I'll just put 7 where b is and then we'll have plus or minus the square root, we put 7 in again because 7 is b.
We square that and subtract 4 times a times c, well a is 6 and c is negative 3 and now we just divide that by 2
times a, which is still 6.
And now we just simplify and we're all set.
So here we'll get negative 7 plus or minus the square root, 7 squared is 49, 4 times 6 times negative 3 comes
out to 72 uh you have these negatives there that comes out to positive 72 at the end and we divide that by 12.
And again you just simplify and when you get done with this you should have, in this case, something pretty nice.
As it turns out, (negative?) 49 plus 72 is 121and that's actually a nice square root.
That is actually 11 right there, the square root of 121 is 11 and so to finish this off we'll have negative 7 plus or
minus 11 over 12.
Now there is one last step that we need to do here, plus or minus means you have two answers.
One where it's a plus and where it's a minus, we need to find both.
Well negative 7 plus 11 over 12 works out to 4/12 which is 1/3.
Negative 7 minus 11 over 12 is negative 18/12 which reduces to negative 3/2.
So we have two answers here, 1/3 and negative 3/2.
Now factoring would have worked on this one, example one actually turns out to factor into 3x minus 1 times 2x
plus 3 and that's wonderful, you could have done that instead.
The problem is that the factoring doesn't work on example two so let's take a look at example two.
A equals negative 5, b equals negative 1, and c equals 2.
Here we just substitute everything into the quadratic formula yet agian.
We'll have x equals the negative of b which is negative 1, be careful that also might come out positive cause we
have two minus' there.
Then we'll have negative 1 squared minus 4 times negative 5 times 2 and we divide all of that by 2 times
negative 5.
Okay, so we'll have 1 plus or minus the square root, negative 1 squared is positive 1, this comes out to a total
of 40 and we divide that whole thing by negative 10, make sure that the main fraction bar covers the entire
top, don't just put it under the root, put it under the entire top.
It is different if you don't put it under the entire top, anyway 1 plus 40 is obviously 41.
Oh this is negative 10 down there so we have 1 plus or minus the square root of 41 over negative 10.
We don't really like having a negative denominators though because it makes simplifying a little harder then
it has to be.
What I am going to do is multiply both the top and the bottom by negative 1, now remember when you do that
you've got to do that to all three terms. All three terms!
1 times negative 1 is obviously negative 1, the fact that this is plus or minus though means the sign won't really
change cause if it's negative it becomes positive, if it's positive it becomes negative.
Not really any need to fuss with that because you get two answers either way and it will be the same two
answers and then on the bottom we have 10 and now we're done.
So now we have negative 1 plus or minus the square root of 41 over 10 and you can find exactly what that is in
your calculator but I wouldn't worry about it.
You can see here that factoring wouldn't have worked here because that answer is much uglier than what we
would have gotten with factoring.
So practice the quadratic formula but you will need to memorize it but with a little bit of practice you'll be
surprised, it's not that hard.
