>> Use the quadratic formula to solve
x squared minus 2x minus 4 equals 0.
We know that the quadratic formula is
given by x equals the opposite of b plus
or minus the square root of b
squared minus 4ac all over 2a.
Now, the first thing that we need to do is to
figure out what the values of a, b and c are.
Remember that a is the coefficient on x
squared, in this case, that's going to be 1,
b is the coefficient on the x term,
in this case, that's negative 2,
and c is the constant term, in
this case, that is negative 4.
Never forget to include the negative sign when
one of these coefficients is a negative number
or we'll end up with the wrong answer.
Now, let's go and plug these quantities in.
We start off with the opposite
of b. Now, b is negative 2.
The opposite of negative 2 is positive 2.
Then, we have plus or minus the square
root of - first of all, we have b squared.
Negative 2 squared is positive 4.
So, we're going to start off with 4, and then
minus 4 times a, which in this case is 1,
times c, which in this case is negative 4,
all over 2 times a, which in this case is 1.
In general, when I'm solving
problems with a quadratic formula,
I will take the b and square it right away.
In this case, b squared was 4,
and always, this will be positive.
We cannot square a number
and end up with a negative.
So, this is always going to be positive.
But then I don't evaluate
the negative 4ac all at once.
I just write negative 4 times a times c,
because if I evaluate all of this in my head,
I'm more likely to make a mistake.
So, I usually just write in the values of a and
c and then multiply them out in the next step,
which we're going to do right now.
So, we've got 2 plus or minus
the square root of 4.
And, then in the second term, it's
going to end up being positive,
because we're subtracting
something with a negative factor.
So, those negatives will
cancel and turn into a plus.
And, 4 times 1 times 4 is 16.
So, we're going to have 4 plus
16 underneath that radical.
In the denominator, 2 times 1 is 2.
Now, let's keep evaluating under that radical.
4 plus 16 is 20, so that's
going to become a root 20.
And, 20 isn't a perfect square, but
we can simplify it by factoring it
so that the first factor is a perfect square.
In this case, 20 is 4 times 5.
And, we can evaluate the square root of 4.
The square root of 4 is 2.
So, this will become 2 plus
or minus 2 root 5 all over 2.
I'm going to rewrite this up here,
2 plus or minus 2 root 5 over 2.
Now, at this point, we can split up those
two terms so that this is two fractions.
We've got 2 over 2 plus or
minus 2 root 5 over 2.
And, the reason we want to split it up is
so that we can simplify each
of the fractions on their own.
In this case, 2 divided by 2 is 1, and then plus
or minus 2 root 5 divided by 2 is just root 5.
The 2's cancel.
And so, for this problem, our solution is x
equals 1 plus or minus the square root of 5.
And, that's our final answer.
