1. Solving Radical Equations

2. How Do We Solve Radical Equations?
Do Now: Simplify the given expression.  

3. How do we solve rational expressions?
Isolate radical if necessary.
Get rid of radical by raising both sides of the equation to the inverse power of the root
Solve using inverse operations

4. An equation in which a variable occurs in the radicand
Radical Equations
An equation in which a variable occurs in the radicand
is called a radical equation. It should be noted, that
when solving a radical equation algebraically,
extraneous roots may be introduced when both sides of
an equation are squared. Therefore, you must check
your solutions for a radical equation.
Check:
L.S R.S.
Solve: √ x = 0
x ≥ 3
√ x
√ x - 3 = 3
(√ x - 3 )2 = (3)2 √
3 - 3
x - 3 = 9
x = 12
Therefore, the solution
is x = 12.

5. Example 1 Isolate radical if necessary. Not necessary in this problem
2. Get rid of radical using inverse operations
3. Now just solve using inverse operations
Wait!!!!
You are not done!!!!
You must check for extraneous solutions.

6. Extraneous Solutions: Are solutions you get with correct math that give you a domain issue when you plug them into the original functions or give you a false statement when you plug them into the original function.
1. A negative under an even radical.
2. A zero in a denominator.
Check Answer(s):
1. Take original problem
2. Insert answer into problem.
3. Check to see if you get a true statement.
True statement so -2 is a solution.

7. Y’all Try Some 

8. Example 2
Now it is time to Check:

9. Example 3
Now Check: (0)
Now Check: (3)

10. Try Some 

11. Example 4
The amount of power used by an appliance is given by where I is the current (in amps), R is the resistance (in ohms), and P is the power (in watts). If the appliance uses I=12 amps, and R=25 ohms, find the power P.

12. 010607b RADICALS: Solving Radicals12

14. 080602b RADICALS: Solving Radicals14

16. ≠ Solving Radical Equations 4 + √ 4 + x2 = x Check: x √ 4 + x2 = x - 4
Since
the solution of
is extraneous. Therefore,
there are no real roots.
x = ≠

17. 080104b RADICALS: Solving Radicals17

19. 010305x RADICALS: Solving Radicals19

20. x = -1 is an extraneous solution.

21. Solving Radical Equations
Solve
x ≥ -2
Set up the equation so that
there will be one radical on
each side of the equal sign.
Square both sides.
2x + 4 = x + 7
x = 3
Simplify.
L.S R.S.
Verify your solution.
Therefore, the
solution is
x = 3.

22. Squaring a Binomial (a + 2)2 = a2 + 4a + 4 ( 5 + √x - 2 )2 (a√x + b)2
Note that the middle term is
twice the product of the two
terms of the binomial.
(a + 2)2 = a2 + 4a + 4
( 5 + √x - 2 )2
The middle term will
be twice the product
of the two terms.
A final concept that you
should know:
(a√x + b)2
= a2(x + b)
= a2x + ab

23. Solving Radical Equations
Set up the equation
so that there will be
only one radical on
each side of the equal sign.
Solve
Square both sides
of the equation.
Use Foil.
Simplify.
Simplify by dividing
by a common factor of 2.
Square both sides of the
equation.
Use Foil.

24. x - 3 = 0 or x - 7 = 0 x = 3 or x = 7 Solving Radical Equations
Distribute the 4.
Simplify.
Factor the quadratic.
Solve for x.
x - 3 = 0 or x - 7 = 0
x = 3 or x = 7
Verify both solutions.
L.S R.S.
L.S R.S.

25. You must square the whole side NOT each term.
One more to see another extraneous solution:
a solution that you find algebraically but DOES NOT make a true statement when you substitute it back into the equation.
The radical is already isolated 2 2
Square both sides
You must square the whole side NOT each term.
This must be FOILed
You MUST check these answers
Since you have a quadratic equation (has an x2 term) get everything on one side = 0 and see if you can factor this
Doesn't work! Extraneous
It checks!

26. Let's try another one: First isolate the radical - 1 - 1
Remember that the 1/3 power means the same thing as a cube root.
- 1
- 1
Now since it is a 1/3 power this means the same as a cube root so cube both sides 3 3
Now solve for x
- 1
- 1
Let's check this answer
It checks!

27. Graph Graphing a Radical Function The domain is x > -2.
The range is y > 0.

28. Solving a Radical Equation Graphically
The solution will be the
intersection of the graph
Solve
and the graph of
y = 0.
The solution
is x = 12.
Check:
L.S R.S.

29. Solving a Radical Equation Graphically
Solve
The solution is
x = 3 or x = 7.

30. Solving Radical Inequalities
Solve
Find the values for which
the graph of
Note the radical
7x - 3 is defined only
when
is above the graph of
y = 3.
The graphs intersect at
x = 4.
x > 4
Therefore, the solution
is x > 4.

31. Solving Radical Inequalities
Solve
x > -1
The graphs intersect
at the point where
x = 8.
x ≥ -1 and x < 8
The solution is
-1 < x and x < 8.