- [Instructor] So solving
quadratic equations
by the quadratic formula.
So if we have a quadratic equation,
which is a x squared, plus
b x plus c equals zero,
and I cannot factor it,
and I cannot use a square root property,
then what I would use is
the quadratic formula.
So the quadratic formula calculates
my two answers, using the standard form,
so written out as a x squared
plus b x plus c equals zero.
It's gonna use coefficients a, b and c.
So if we have the standard form,
then x equals, so here's the formula,
it's gonna be a big fraction,
negative b plus or minus the square root
of b squared minus 4 a c, over 2 times a.
So by plugging in a, b,
and c, the coefficients,
I can calculate what x is.
So for example, if I
have that I wanna solve
x squared plus 7 x equals 3.
Okay, so first off, it's
not in standard form yet.
So first I'm going to
subtract 3 on both sides
so that I have this equal to zero.
Now I'm gonna list out, my a is 1.
B is 7, and c is -3.
And c is -3.
So if I take that formula, and I start
plugging my numbers in,
so I've got negative b, so -7,
plus or minus the square
root of b squared,
so 49, 7 times 7, 49,
minus 4 times 1 times -3, 4, a, c,
over 2 times a, 2 times 1.
So -7 plus or minus
let's see, I'm just gonna
write out the bottom as 2.
Now inside the parenthesis,
so I already did 7 squared is 49.
If I do 4 times 1 times -3,
so that's gonna be -12.
Sometimes it too helps, just kind of
jot over to the side, the
work under the square root.
So that's gonna become a plus 12
so square root of, that's gonna be 61.
So the square root of 61.
Now, can't really simplify
much more right here,
so let's actually do
the square root of 61,
get an approximation.
Square root of 61 is about 7.81, over 2.
So this is two different
answers, that plus or minus.
This means -7 plus 7.81, divided by 2,
and -7 minus 7.81, all divided by 2.
So when I calculate that,
if you're gonna put this all
on one step in your calculator,
you can do this all, just make sure
you would have to do parenthesis on top
in the numerator, so that it does
the -7 plus 7.81 before
dividing the whole thing by 2.
So for my addition problem, I'll have 0.41
and for the subtraction, -7 minus 7.81
divided by 2, -7.41.
So those are the two values that x can be.
If I plug it back into
the original equation,
0.41 squared, plus 7 times 0.41,
I should get about 3.
If I do the same thing with -7.41,
- 7.41 squared, plus 7 times
- 7.41 should be approximately 3.
Okay, so those are my two answers.
Okay, let's do one more.
So this one, we are going to solve,
we're gonna solve x squared minus 8 x
minus 9 equals zero.
So this one's already set up nice for us.
A is one, b is -8, and c is -9.
So back to my formula.
X is negative b, so I have negative, -8
plus or minus square root of b squared,
- 8 squared, minus 4, times a, times c,
so -4 times 1 times -9,
over 2 times 1.
So this one, when b is negative,
be careful of the signs,
negative negative, that's
gonna go back to positive 8,
plus or minus, inside the parenthesis,
- 8 squared, that's a positive 64,
- 4 times 1 times -9,
that's gonna be a plus 36,
so that's a square root of 100 over 2.
So 8 plus or minus,
square root of 100 is 10,
over 2, and it's always best,
write it out separately.
8 plus 10 divided by 2,
8 minus 10 divided by 2.
So 8 plus 10, 18, divided by 2 is 9.
8 minus 10 is -2,
divided by 2 is -1.
So x is 9 and -1.
So notice, this one
came out to nice answers
instead of having to use our calculator
to do a square root that doesn't come out.
What I could have actually
done with this problem,
let's switch this up, let
me go back to the start.
X squared minus 8 x minus 9 is zero.
I could have actually factored this one.
If I did 1 times 9 minus and plus,
minus 9 and plus 1, if
I factored that out,
I'd get back to x squared
minus 8 x minus 9.
So by factoring, my answers would be
negative 1 and 9, same thing,
just different strategy.
So actually factoring would
have been a little quicker.
But when this is one that
I could have factored,
that square root part's
gonna come out nice,
so I can simplify this.
But either way, if I didn't notice,
still get the same answer,
using the quadratic formula.
