(male narrator)
In this video,
we will look at how we
can use the quadratic formula
in order
to solve equations.
This quadratic equation:
ax squared, plus bx,
plus c, equals 0;
we found, by completing
the square, has the solution:
x is equal
to the opposite of b,
plus or minus
the square root of b squared,
minus 4ac,
all over 2a.
By committing this formula
to memory,
we can quickly solve
quadratic equations.
For example,
in this problem,
we see it set up
exactly like the other equation,
where the a is 6, the b is 7,
and the c is -3.
Notice the sign goes
with the number on the 3.
If x is equal
to the opposite of b,
plus or minus the square root
of b squared,
minus 4ac,
all over 2a,
we simply have to plug
these values into the formula:
x is equal
to the opposite of b;
b is 7,
so we have -7;
plus or minus the square root
of b squared, or 7 squared;
minus 4a,
which we said was 6;
c, which we said was -3;
all over 2a,
which we said was 6.
We simply have
to evaluate this
by using
the order of operations.
We can do the exponents
and all the multiplication
at the same time
without violating
order of operations,
because it does not get
in the way
of any
higher-ordered operations.
This gives us
x equals -7;
plus or minus the square root
of 7 squared, which is 49;
plus, because we've got
two negatives--
negative times a negative's
a positive;
4 times 6, times 7, is 72;
all over 2 times 6,
which is 12.
Now, we can add
inside the radical,
and x is equal to -7,
plus or minus the square root
of 121, over 12.
The square root of 121 is 11,
and so, we have -7,
plus or minus 11, over 12.
We can consider
the two possibilities
of the positive and negative
to simplify:
-7 plus 11 is 4, over 12;
-7 minus 11 is -18, over 12.
Both of those reduce
to give our final solutions:
1/3 and -3/2.
Let's try another problem
where we use
the quadratic formula
in order to solve for x.
In this problem, we can see
that a is 5, b is -1--
there's always a 1 in front
of a variable--and c is 2.
If the quadratic formula
tells us
that x is equal
to the opposite of b,
plus or minus
the square root of b squared,
minus 4ac,
all over 2a,
then we simply have
to plug these values in.
The opposite of b--
the opposite of -1 is +1--
plus or minus the square root
of b squared, or -1 squared;
minus 4a, which is 5;
c, which is 2;
all over 2a,
which is 5.
We'll simplify the exponents
and multiplication first.
So we have
x is equal to 1,
plus or minus
the square root of 1,
minus 40...
all over 10.
Doing the subtraction,
we get x is equal to 1,
plus or minus the square root
of -39, over 10.
We will want
to simplify the radical,
and the only thing
we can simplify on this radical
is we can pull
the negative out.
The square root
of a negative is i.
For our final answer,
x is equal to 1,
plus or minus i,
times the square root of 39,
over 10.
By plugging our numbers in
for a, b, and c
into the quadratic formula,
we could solve a problem
that would otherwise
be quite difficult
by completing
the square.
