(male narrator)
In this video,
we will look at using
the quadratic formula
where we may have
to do some work to set it up.
Before using
the quadratic formula,
it is very important that
the equation must...equal 0.
So let's take a look
at some examples
where it doesn't equal 0
to begin with.
When we're solving, we like
the x squared to be positive,
so let's move everything
in this equation
to the right side
to make it equal 0.
We can do this
by adding 7x...
and subtracting 15.
When we do,
because there's no like terms,
we get 2x squared, plus 7x,
minus 15, equals 0.
Now, we're ready to use
the quadratic formula,
where a is 2, b is 7,
and c is -15.
Plugging our values
into the quadratic formula:
x is equal to the opposite of b,
or -7;
plus or minus the square root
of b, or 7 squared;
minus 4a, or 2;
times c, which is -15;
all over 2a,
which is 2.
We can now simplify
by doing the multiplication
in the exponents
to get x equals -7;
plus or minus
the square root of 49;
plus, because we have
a negative times a negative;
4 times 2, times -15, is 120;
all over 2 times 2, which is 4.
Adding inside the radical,
we find out x is equal to -7,
plus or minus
the square root of 169, over 4.
The square root of 169 is 13,
so we have -7,
plus or minus 13, over 4.
We can find
our two solutions
by splitting
into the two equations
where we add and subtract:
-7 plus 13 is 6, over 4;
and -7 minus 13 is -20, over 4.
These will reduce to our
final answers: 3/2 and -5.
Let's take a look
at one more example
where we have to make
the equation equal to 0 first
before we can solve
using the quadratic formula.
In this problem,
it equals 7.
We need to move
the 7 over
by subtracting 7
from both sides,
lining up like terms,
to get 3x squared, plus 5x,
minus 5, equals 0.
We can now look
at this
and identify
what a, b, and c are
in order to plug them
into the quadratic formula.
The quadratic formula says
x is equal to the opposite of b,
or -5;
plus or minus the square root
of b squared, 5 squared;
minus 4a, which is 3;
c, which is -5;
all over 2a,
which is 3.
Start simplifying
by doing the multiplication
in the exponents
to get x equals -5,
plus or minus
the square root of 25,
plus 60,
all over 6.
Adding inside the radical
tells us that x is equal to -5,
plus or minus
the square root of 85,
all over 6.
In this case, we would want
to check to see
if we can simplify
that radical.
However, finding
the prime factorization of 85,
by dividing by 5,
we find out is 17,
and 17, and 1.
We can't simplify
5 times 17,
and so, this radical
and expression
is completely simplified.
Our final solution
is x equals -5,
plus or minus
the square root of 85,
over 6.
By making
the equation equal to 0
and plugging in our numbers
for a, b, and c
into the quadratic formula,
we can quickly arrive
at our solution.
