1. Exercise
Solve x – 14 = 35.
x = 49

2. Exercise
Solve 6x = 108.
x = 18

3. Exercise
Solve 9x + 14 = – 49.
x = – 7

4. Exercise
Solve 4x – 81 = 99.
x = 45

5. Exercise
Solve 6(3x – 13) = 84.
x = 9

6. Radical Equation
A radical equation is an equation that contains a variable in a radicand.
Example: √ x = 5

7. Remember: 62 = 36, so √ 36 = 6.
In general, (√ x )2 = x and
(√ x + 3 )2 = x + 3.

8. When solving, be sure to square both sides of the radical equation.

9. Example 1
Solve √ x = 5.
√ x = 5
√ x = 5
(√ x )2 = 52
√ 25 = 5
x = 25

10. There is no number that makes this equation true.
Example 2
Solve and check √ x = – 8.
√ x = – 8
√ x = – 8
(√ x )2 = (– 8)2
√ 64 = – 8
x = 64
8 = – 8; false
There is no number that makes this equation true.

11. Solving Radical Equations
Isolate the radical on one side of the equation.
2. Square both sides of the equation.
3. Solve the resulting equation, if necessary.
4. Check the solution.

12. Example 3 Solve and check √ 3x + 2 = 8. √ 3x + 2 = 8

13. Example 3 Solve and check √ 3x + 2 = 8. x = 12 √ 3(12) + 2 = 8
√ = 8
6 + 2 = 8
8 = 8

14. Example 4 Solve and check √ 10 + x + 6 = 7. √ 10 + x + 6 = 7

15. Example 4 Solve and check √ 10 + x + 6 = 7. 10 + x = 1
√ 10 + (– 9) + 6 = 7
1 + 6 = 7
√ = 7
7 = 7

16. The equation has no real solution.
Example 5
Solve – 4 √ x = 28.
– 4 √ x = 28
– 4
√ x = – 7
The equation has no real solution.

17. Example
Solve √ x = 6.
x = 36

18. Example
Solve √ x + 3 = 8.
x = 61

19. Example
Solve √ x + 3 = 8.
x = 25

20. Example
Solve 2 √ x = 8.
x = 16

21. Example
Solve √ 2x = 8.
x = 32

22. Example
Solve √ 3x – 1 = 2.
x =
5 3

23. Example
Solve √ 1 – x = 4.
x = – 15

24. Example
Solve 4 √ 2x – = 22.
x =
21 2

25. Example
Solve √ 2x + 1 = √ x + 5.
x = 4

26. Exercise
The square root of a number, decreased by 2, is 4. Find the number.
n = 36

27. Exercise
The square root of the result of decreasing a number by 2 is 4. Find the number.
n = 18

28. Exercise
Seven more than the square root of a number is 12. Find the number.
n = 25

29. Exercise
The square root of 7 more than a number is 12. Find the number.
n = 137