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How Do We Solve Radical How Do We Solve Radical Equations?Equations?
• Do Now: Do Now: Simplify the given Simplify the given expression.expression.
1. 2. 1. 2.
An equation in which a variable occurs in the radicandis called a radical equation. It should be noted, that when solving a radical equation algebraically, extraneous roots may be introduced when both sides ofan equation are squared. Therefore, you must check your solutions for a radical equation.
Solve: √ x - 3 - 3 = 0
√ x - 3 = 3
(√ x - 3 )2 = (3)2
x - 3 = 9 x = 12
Check:
√ x - 3 - 3
√ 12 - 3 - 33 - 3 0
0
Therefore, the solution is x = 12.
x ≥ 3
Radical Equations
L.S. R.S.
If
,5212 x
then x is equal to (1) 1 (3) 5 (2) 2 (4) 4
5
102
912
312
5212
x
x
x
x
x
What is the solution of the equation
?6332 x
(1) 42 (3) 3 (2) 39 (4) 6
42
842
8132
932
6332
x
x
x
x
x
4 + √ 4 + x2 = x
√ 4 + x2 = x - 4
4 + x2 = x2 - 8x + 16 8x = 12
x
3
2
3
2
13
2,Since the solution of
x = 3
2 is extraneous. Therefore,
there are no real roots.
Check: 4 4 x2
4 4 3
2
2
4 4 9
4
4 25
4
4 5
213
23
2≠
(√ 4 + x2)2 = (x - 4)2
x 3
2
Solving Radical Equations
The solution set of the equation
x x 6
is
(1) {–2,3} (3) {3} (2) {–2} (4) { }
x x
x x
x x
x x
x x
6
6
6 0
3 2 0
3 2
2
2
( )( )
What is the solution set of the equation
9 10x x ? (1) {-1} (3) {10} (2) {9} (4) {10, -1}
9 10
9 10
9 10 0
10 1 0
10 1
2
2
x x
x x
x x
x x
x x
( )( )
x = -1 is an extraneous solution.
2x 4 x 7 0.
Set up the equation so thatthere will be one radical oneach side of the equal sign.
2x 4 x 7
Square both sides.2x 4 2 x 7 2
Simplify. 2x + 4 = x + 7 x = 3
Verify your solution.
2x 4 x 7 0Therefore, thesolution isx = 3.
x ≥ -2Solve
Solving Radical Equations
2(3) 4 3 7
10 10
0
L.S. R.S.
Squaring a Binomial
(a + 2)2 = a2 + 4a + 4Note that the middle term is twice the product of the two terms of the binomial.
(a√x + b)2
( 5 + √x - 2 )2
The middle term will be twice the product of the two terms.
5 x 2 2
10 x 2
5 x 2 2 A final concept that you should know:
25 10 x 2 (x 2)
x 23 10 x 2 = a2x + ab= a2(x + b)
5x 1 3x 5 2.Set up the equationso that there will beonly one radical oneach side of the equal sign.
Square both sidesof the equation.
Simplify.
Simplify by dividingby a common factor of 2.
Square both sides of the equation.
5x 1 2 2 3x 5 2
5x 1
5x 1 3x 1 4 3x 5
2x 2 4 3x 5
x 1 2 3x 5
x 1 2 2 3x 5 2x2 2x 1 4(3x 5) Use Foil.
5x 1 2 3x 5
Use Foil.4 4 3x 5 (3x 5)
Solve
Solving Radical Equations
x2 2x 1 4(3x 5) Distribute the 4.
x2 2x 1 12x 20 Simplify.
x2 10x 21 0 Factor the quadratic.
(x 3)(x 7) 0Solve for x.
x - 3 = 0 or x - 7 = 0 x = 3 or x = 7 Verify both solutions.
5x 1 3x 5 2
5(3) 1 3(3) 5
4 2
2
5x 1 3x 5 2
5(7) 1 3(7) 5
6 4
2
Solving Radical Equations
L.S. R.S. L.S. R.S.
One more to see another extraneous solution:
313 xx The radical is already isolated
2 2You must square the whole side
NOT each term.
9613 2 xxx
Square both sides
Since you have a quadratic equation (has an x2 term) get everything on one side = 0 and see if you can factor this
1,8 xx
You MUST check these answers
55
38183
313 xxThis must be FOILed
0892 xx
018 xx 22
31113
Doesn't work!Extraneous
It checks!
a solution that you find algebraically but DOES NOT make a true statement when you substitute it back into the equation.
Let's try another one:
0112 3
1
x First isolate the radical
- 1 - 1
112 3
1
x3 3
112 x
Now since it is a 1/3 power this means the same as a cube root so cube both sides
Now solve for x
- 1 - 1
22 x1x
Let's check this answer
011123 00 It checks!
Remember that the 1/3 power means the same thing as a cube root.
y x 2.Graph
The domain is x > -2.The range is y > 0.
Graphing a Radical Function
Solve x 3 3 0. The solution will be theintersection of the graph
y x 3 3
and the graph ofy = 0.
The solutionis x = 12.
Check:
x 3 3 0
12 3 39 33 3
Solving a Radical Equation Graphically
L.S. R.S.
5x 1 3x 5 2.
The solution isx = 3 or x = 7.
Solve
Solving a Radical Equation Graphically
7x 3 2 3.Find the values for which the graph of
y 7x 3 2
is above the graph of y = 3.
The graphs intersect at x = 4.
Note the radical7x - 3 is defined only
when .
Therefore, the solutionis x > 4.
x > 4
Solving Radical Inequalities
Solve
x 3
7
x 1 3.
The solution is -1 < x and x < 8.
The graphs intersectat the point where x = 8.
x ≥ -1 and x < 8
x > -1
Solving Radical Inequalities
Solve
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