Math 20-1 Chapter 5 Radical Expressions and Equations

5.3 Solve Radical Equations

Teacher Notes

5.3 Radical Equations• A Radical Equation must have at least one

radicand containing a variable

• The Power Rule: Keeping Equality Balanced• If we raise two equal quantities to the same power,

the results are also two equal quantities.• If x = y then x2 = y2

• Warning: These are NOT equivalent Equations!

221743 3 3 xxxxx

The values for the variable in the radicand must be considered when solving. 5.3.1

• Start with a simple original equation:• x = 3• Square both sides to get a new equation:• x2 = 32 which simplifies to x2 = 9• The only solution to x = 3 is 3• x2 = 9 has two solutions 3 and -3• -3 is considered extraneous

Extraneous RootsAn extraneous root is a number obtained in solving an equation that does not satisfy the initial restrictions on the variable.

5.3.2

Equations Containing One Radical

Isolate the radical term on one side of the equation and then apply the Power Rule with squares.

Determine the roots of the radical equation:

3 4x

2 2

3 4x

3 16x 13x

Verify by substitut

3 4

16 4

4

ion

13

:

4

a) 3 4x

Verify the radicand restrictions

3 0

3x

x

5.3.3

Equations Containing One Radicalb) 2 4x x

2242 xx

28162 xxx 29140 xx

1490 2 xx)2)(7(0 xx

7, 2x

Verify by sub :

7

7 72 4

9 3

3 3

x

2 4

4 2

2

2 2

2 2

x

Therefore the solution is x = 2.

5.3.4

Why would just verifying the radicand restrictions not be sufficient in this case?

Your TurnAlgebraically determine the roots of

Equations Containing One Radical

xx 4330

4330 xx

224330 xx

1450 2 xx

)2)(7(0 xx

7, 2x

3

V

0

erify by sub:

7

73( ) 4

9

3 3

7

3

x

30 3( 2

2

) 4

36 6

6 6

2

x

Therefore the solution is x = 7.5.3.5

Can you just verify the radicandrestrictions?

Equations Containing Two Radicals• Separate the radicals: one on each side of the equality sign• Square both sides of the equation, not individual terms• If a radical is still present, isolate that radical and square both sides a second time

Algebraically determine the roots

a) 4 60x x

2 2

4 60x x

224 60x x

16 60x x

15 60x

4x

4 60x x

4 4 64 0

4 2 64

8 8

Verify by sub:

4x

5.3.6

Verify the radicand restrictions

Equations Containing Two RadicalsSolve 2 3 7 2x x

2 3 7 2x x

2 3 2 7x x

2 2

2 3 2 7x x

2 2

2 3 2 7x x

2 3 4 4 7 7x x x

2 3 4 7 11x x x 14 4 7x x

2214 4 7x x

Verify the radicand restrictions.

5.3.7

2 28 196 16( 7)x x x 2 28 196 16 112x x x

2 44 84 0x x

( 42)( 2) 0x x

42 0 or - 2 0x x or 42 2 x x

Equations Containing Two Radicals

:

42

Check

x

2( ) 3 7 2

81 49

42 4

2

9 7 2

2

2

2

:

2

Check

x

2( ) 3 7 2

1

2 2

9 2

1 3 2

2 2

Therefore x = 42

5.3.8

Your Turn

Solve:

Equations Containing Two Radicals

2 3 2 2x x

2 3 2 2x x

2 3 2 2x x

2 2

2 3 2 2x x

2 3 4 4 2 2x x x

3 4 2x x

223 4 2x x

2 6 9 16( 2)x x x

5.3.9

2 6 9 16 32x x x 2 22 23 0x x

( 23)( 1) 0x x or 23 1x x

Equations Containing Two Radicals

:

23

Check

x

2( ) 3 2 2

49 25

23 2

2

7 5 2

2

3

2

:

1

Check

x

2( ) 3 2 2

1

1

1

1

2

0 2

Therefore x = 23

5.3.10

AssignmentSuggested Questions:

Page 300:2, 3a,b, 4a,b, 5, 6b, 7a, 8a,c, 9d, 10a,d12, 13, 15,

5.3.11