1. Radical
Equations

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

3. 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.

4. 010607b RADICALS: Solving Radicals4

6. 080602b RADICALS: Solving Radicals

8. ≠ 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 = ≠

9. 080104b RADICALS: Solving Radicals9

11. 010305x RADICALS: Solving Radicals11

12. x = -1 is an extraneous solution.

13. 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.

14. 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

15. 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.

16. 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.

17. 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!

18. 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!

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

20. 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.

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

22. 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.

23. 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.

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