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(male narrator) In this video,
we will solve equations by completing the square
that might give us irrational or complex solution.
If we can't simplify the radical completely,
we will simply simplify...
what we can.
For example, in this problem,
if we were to solve by completing the square,
we first separate the variables from the numbers
by adding 3 to both sides.
We now have 2x squared, minus 8x, equals 3.
Our next step is always to get the x squared alone
by dividing by a, or the number in front.
Dividing everything by 2
gives us x squared, minus 4x, equals 3/2.
Now, we're ready to complete the square by finding the c,
which is 1/2 times b-- or -4 in this case--squared:
1/2 of -4 is -2 squared, which is 4.
This is what we want to add to both sides of the equation.
When we do, the left side will always be a perfect square.
Square root of the first term,
sign from the middle, square root of the last term:
x minus 2 squared equals...
On the right side, we'll need a common denominator,
which we can get by multiplying by 2:
3/2 plus 8/2 gives us 11/2.
We can now get rid of the square
by taking the square root of both sides.
On the right side, you'll notice we have a square root of 2
and the denominator, but also under a radical.
We'll need to rationalize this denominator
by multiplying by root 2 on top and bottom.
This gives us the square root of 22/2.
Don't forget the plus or minus, square root 22/2.
To get the x alone, we add 2 to both sides.
And we get x equals 2,
plus or minus root 22/2.
Let's get a common denominator
by multiplying
the top and bottom by 2:
2 times 2 is 4, plus or minus the square root of 22,
over the common denominator of 2.
Let's try another example
where the square root doesn't come out evenly,
and we simply simplify what we can.
In this problem,
we'll start by subtracting 2 from both sides,
separating the variables from the numbers.
Next, we must divide by a,
so the x squared has a known number in front of it.
Remember, we divide every term by 5,
so we get x squared, minus 3/5x, equals -2/5.
We can now find c by taking 1/2 of b, or the -3/5 squared.
This is -3/10 squared, or 9/100.
By adding 9/100 to both sides, we've completed the square.
On the left side, it will factor to a perfect square.
The square root of the first term,
sign from the middle,
and the square root of the last term.
On the right side, we need to multiply by 20
to get a common denominator:
-80/100, plus 9/100...whoops, 2 times 20 is 40...-40/100.
This is gonna give us -31/100.
We're now ready to take the square root of both sides,
so that square and square root undo each other,
and we get x minus 3/10 equals;
because there's a negative,
it will come out as an i root 31;
square root of 100 is 10.
Don't forget the plus or minus
whenever we take the square root of both sides.
To get x alone, we just have to add 3/10,
and because we already have
a common denominator,
we can write
our final answer as 3,
plus or minus i root 31,
all over the common denominator
of 10.
Simplify what we can when we can't take the root.