Example 1: Solve Information for Solving Radical Equations
To solve most equations, we usually isolate x. Can we add 4 to both sides here?
We could, but it wouldn't help because it would be outside the square root sign on the left.
What can we do?
Raise both sides to the same power.
What power is helpful here?
The second power, or we need to square both sides.
Do it.
, or
7x - 4 = 9
Now can we add 4 to both sides?
Yes, this will help because the 4 on the left is no longer under a square root sign.
Do it.
7x = 13
Solve for x
Let's check this in the original equation.
We get 
3 = 3
.Information for Solving Radical Equations
This is different from Example 1 in that we have a multiple of a radical on one side. How can we isolate the radical?
We can divide both sides by 3.
Do it.
We get 
.
Now square both sides.
, or
Solve for x.
Adding 5 to both sides, we get
Check this result in the original equation.
We get
OK
.Information for Solving Radical Equations
What do we do first?
Isolate the radical.
Do it.
Subtracting 7 from both sides, we get
Does this make sense?
No. The radical sign means the positive square root, and yet the right-hand side is negative.
What can we conclude?
We conclude that there is no solution to this problem.
Information for Solving Radical Equations
Let's see if squaring both sides without isolating the radical simplifies this, as it did in Example 1 and 3. Set it up.
We get 
Do the squaring.
, or
, or
, or
Notice that we didn't get rid of the radical. How is this problem different from Example 1?
In Example 1, the radical was already isolated. Here it was not, and you can see why it is necessary to do so.
How can we do that here?
By subtracting 6 from both sides.
Do it.
What do we do next?
Divide both sides by 5.
Do it.
Now what do we do?
Square both sides.
Do it.
, or
Now add 3 to both sides.
Divide by 2 on both sides.
Check this result in the original equation.
6 + 5 * 0.8 ? 10, or
6 + 4 ? 10, or
10 = 10
.Information for Solving Radical Equations
This problem is different in having two radicals. How can we isolate them?
We can add 
to each side.
Do it.
We get
Can we still square both sides of the equation?
Yes.
Do it.
, or
2x + 3 = 5x + 1
Solve for x.
Subtracting 3 from both sides, we get
2x = 5x + 1 - 3 = 5x -2
Subtracting 5x from both sides, we get
2x - 5x = -2, or
-3x = -2
Dividing by -3 on both sides, we get
Check this solution in the original equation.
, or
, or
Information for Solving Radical Equations
How can we proceed?
First, isolate 
How?
Subtract 2.7 from both sides.
Do it.
Divide by 5 on both sides.
Now what can we do?
We want q, not its square root, so let's square both sides.
Do it.
, or
q = (3.1)2 = 9.61
Check this value in the original equation.
We get
15.5 + 2.7 ? 18.2
18.2 = 18.2
How do we proceed?
We need to raise each side to a power so that we isolate x.
What power do we need to use here?
The negative 4th power.
Do it.
, or
x = 2-4
Try to evaluate 2-4 without a calculator.
Solve for x.
Check this result in the original equation.
(2-4)-1/4 = 2-4*(-1/4) = 2+1 = 2
Let's review functional notation.
Does this require evaluation of f(2)?
No. f(x) = 2 requires finding a value of x which, when substituted in f(x), produces 2.
Does f(x) mean f multiplied by x?
No. f(x) is the name of the set of instructions represented by 
.
So what do we work on?
f
(x) = 2 means to solve
How can we solve this?
By isolating 
Do it.
Add 3 to both sides:
Should we now divide by 8 on both sides?
No, not until we take care of the exponent.
How can we clear the exponent?
By raising each side to the 3rd power.
Do it.
, or
8x = 125
Solve for x.
Check this result in the original equation.
We get
, or
, or
5 - 3 ? 2, or
2 = 2