Radical equations are equations with an unknown variable under a radical sign. To solve radical equations, each side of the equation is squared repeatedly to remove all radicals. This is done because if two expressions are equal, then their squares are also equal. Once all radicals are removed, the resulting equation can be solved normally for the unknown variable. Examples show how to isolate radical terms, expand squared expressions using formulas, and check solutions. Squaring each side must be done carefully to properly isolate radical terms.

1. Radical Equations

2. Radical equations are equations with the unknown x under
the radical.
Radical Equations

3. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
Radical Equations

4. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations

5. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals.

6. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals. Then solve
for x and check the answers.

7. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals. Then solve
for x and check the answers.
Example A. Solve.
a. x = 4

8. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals. Then solve
for x and check the answers.
Example A. Solve.
a. x = 4 square each side

9. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals. Then solve
for x and check the answers.
Example A. Solve.
a. x = 4 square each side
(x)2 = 42

10. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals. Then solve
for x and check the answers.
Example A. Solve.
a. x = 4 square each side
(x)2 = 42
x = 16

11. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals. Then solve
for x and check the answers.
Example A. Solve.
a. x = 4 square each side
(x)2 = 42
x = 16 It works.

12. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals. Then solve
for x and check the answers.
Example A. Solve.
a. x = 4 square each side
(x)2 = 42
x = 16 It works.
b. x – 3 = 4

13. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals. Then solve
for x and check the answers.
Example A. Solve.
a. x = 4 square each side
(x)2 = 42
x = 16 It works.
b. x – 3 = 4 square each side

14. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals. Then solve
for x and check the answers.
Example A. Solve.
a. x = 4 square each side
(x)2 = 42
x = 16 It works.
b. x – 3 = 4 square each side
(x – 3)2 = 42

15. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals. Then solve
for x and check the answers.
Example A. Solve.
a. x = 4 square each side
(x)2 = 42
x = 16 It works.
b. x – 3 = 4 square each side
(x – 3)2 = 42
x – 3 = 16
x = 19

16. Radical equations are equations with the unknown x under
the radical.
To solve radical equations, we use the following fact.
If L=R, then L2 = R2.
Radical Equations
To solve a radical equation, square each side of the equation
(repeatedly if necessary) to remove the radicals. Then solve
for x and check the answers.
Example A. Solve.
a. x = 4 square each side
(x)2 = 42
x = 16 It works.
b. x – 3 = 4 square each side
(x – 3)2 = 42
x – 3 = 16
x = 19 It works.

17. c. 2x + 1 = –3
Radical Equations

18. c. 2x + 1 = –3 square each side
(2x + 1)2 = (–3)2
Radical Equations

19. c. 2x + 1 = –3 square each side
(2x + 1)2 = (–3)2
2x + 1 = 9
Radical Equations

20. c. 2x + 1 = –3 square each side
(2x + 1)2 = (–3)2
2x + 1 = 9
2x = 8
Radical Equations

21. c. 2x + 1 = –3 square each side
(2x + 1)2 = (–3)2
2x + 1 = 9
2x = 8
x = 4
Radical Equations

22. c. 2x + 1 = –3 square each side
(2x + 1)2 = (–3)2
2x + 1 = 9
2x = 8
x = 4
However, x = 4 does not work. So there is no solution.
Radical Equations

23. c. 2x + 1 = –3 square each side
(2x + 1)2 = (–3)2
2x + 1 = 9
2x = 8
x = 4
However, x = 4 does not work. So there is no solution.
Radical Equations
For some problems, we have to square more than once to
eliminate all the radicals.

24. c. 2x + 1 = –3 square each side
(2x + 1)2 = (–3)2
2x + 1 = 9
2x = 8
x = 4
However, x = 4 does not work. So there is no solution.
Radical Equations
For some problems, we have to square more than once to
eliminate all the radicals. Recall the squaring formula that
(A ± B)2  A2 ± 2AB + B2

25. c. 2x + 1 = –3 square each side
(2x + 1)2 = (–3)2
2x + 1 = 9
2x = 8
x = 4
However, x = 4 does not work. So there is no solution.
Radical Equations
Example B. Expand.
a. (x + 4)2
For some problems, we have to square more than once to
eliminate all the radicals. Recall the squaring formula that
(A ± B)2  A2 ± 2AB + B2
Let’s review the algebra below.

26. c. 2x + 1 = –3 square each side
(2x + 1)2 = (–3)2
2x + 1 = 9
2x = 8
x = 4
However, x = 4 does not work. So there is no solution.
Radical Equations
Example B. Expand.
a. (x + 4)2
= (x )2 + 2 * 4 x + 42
For some problems, we have to square more than once to
eliminate all the radicals. Recall the squaring formula that
(A ± B)2  A2 ± 2AB + B2
Let’s review the algebra below.

27. c. 2x + 1 = –3 square each side
(2x + 1)2 = (–3)2
2x + 1 = 9
2x = 8
x = 4
However, x = 4 does not work. So there is no solution.
Radical Equations
Example B. Expand.
a. (x + 4)2
= (x )2 + 2 * 4 x + 42
= x + 8x + 16
For some problems, we have to square more than once to
eliminate all the radicals. Recall the squaring formula that
(A ± B)2  A2 ± 2AB + B2
Let’s review the algebra below.

28. Radical Equations
b. (2x + 1 – 3)2

29. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32

30. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9

31. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1

32. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

33. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

34. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

35. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
x + 2*4 x + 16 = 5x + 4
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

36. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
x + 2*4 x + 16 = 5x + 4 isolate the radical
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

37. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
x + 2*4 x + 16 = 5x + 4 isolate the radical
8x = 4x – 12
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

38. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
x + 2*4 x + 16 = 5x + 4 isolate the radical
8x = 4x – 12 divide by 4
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

39. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
x + 2*4 x + 16 = 5x + 4 isolate the radical
8x = 4x – 12 divide by 4
2x = x – 3
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

40. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
x + 2*4 x + 16 = 5x + 4 isolate the radical
8x = 4x – 12 divide by 4
2x = x – 3 square again
( 2x)2 = (x – 3)2
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

41. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
x + 2*4 x + 16 = 5x + 4 isolate the radical
8x = 4x – 12 divide by 4
2x = x – 3 square again
( 2x)2 = (x – 3)2
4x = x2 – 2*3x + 9
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

42. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
x + 2*4 x + 16 = 5x + 4 isolate the radical
8x = 4x – 12 divide by 4
2x = x – 3 square again
( 2x)2 = (x – 3)2
4x = x2 – 2*3x + 9
0 = x2 – 10x + 9
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

43. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
x + 2*4 x + 16 = 5x + 4 isolate the radical
8x = 4x – 12 divide by 4
2x = x – 3 square again
( 2x)2 = (x – 3)2
4x = x2 – 2*3x + 9
0 = x2 – 10x + 9 0 = (x – 9)(x – 1)
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

44. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
x + 2*4 x + 16 = 5x + 4 isolate the radical
8x = 4x – 12 divide by 4
2x = x – 3 square again
( 2x)2 = (x – 3)2
4x = x2 – 2*3x + 9
0 = x2 – 10x + 9 0 = (x – 9)(x – 1)
x = 9 or x = 1
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

45. Radical Equations
b. (2x + 1 – 3)2
= (2x + 1)2 – 2*32x + 1 + 32
= 2x + 1 – 62x + 1 + 9
= 2x + 10 – 62x+1
Example C. Solve for x.
a. x + 4 = 5x + 4 square both sides
(x + 4)2 = (5x + 4 )2
x + 2*4 x + 16 = 5x + 4 isolate the radical
8x = 4x – 12 divide by 4
2x = x – 3 square again
( 2x)2 = (x – 3)2
4x = x2 – 2*3x + 9
0 = x2 – 10x + 9 0 = (x – 9)(x – 1)
x = 9 or x = 1 Only 9 is good.
When squaring both sides of an equation to remove a radical,
make sure that the radical term is isolated to one side first.

46. b. x + 1 – 3 = x – 8
Radical Equations

47. b. x + 1 – 3 = x – 8 square both sides;
(x + 1 – 3)2 = (x – 8)2
Radical Equations

48. b. x + 1 – 3 = x – 8 square both sides;
(x + 1 – 3)2 = (x – 8)2
x + 1 – 2*3x + 1 + 32 = x – 8
Radical Equations

49. b. x + 1 – 3 = x – 8 square both sides;
(x + 1 – 3)2 = (x – 8)2
x + 1 – 2*3x + 1 + 32 = x – 8
x + 10 – 6x + 1 = x – 8
Radical Equations

50. b. x + 1 – 3 = x – 8 square both sides;
(x + 1 – 3)2 = (x – 8)2
x + 1 – 2*3x + 1 + 32 = x – 8
x + 10 – 6x + 1 = x – 8 isolate the radical
Radical Equations

51. b. x + 1 – 3 = x – 8 square both sides;
(x + 1 – 3)2 = (x – 8)2
x + 1 – 2*3x + 1 + 32 = x – 8
x + 10 – 6x + 1 = x – 8 isolate the radical
10 + 8 = 6x + 1
Radical Equations

52. b. x + 1 – 3 = x – 8 square both sides;
(x + 1 – 3)2 = (x – 8)2
x + 1 – 2*3x + 1 + 32 = x – 8
x + 10 – 6x + 1 = x – 8 isolate the radical
10 + 8 = 6x + 1
18 = 6x + 1
Radical Equations

53. b. x + 1 – 3 = x – 8 square both sides;
(x + 1 – 3)2 = (x – 8)2
x + 1 – 2*3x + 1 + 32 = x – 8
x + 10 – 6x + 1 = x – 8 isolate the radical
10 + 8 = 6x + 1
18 = 6x + 1 div. by 6
3 = x + 1
Radical Equations

54. b. x + 1 – 3 = x – 8 square both sides;
(x + 1 – 3)2 = (x – 8)2
x + 1 – 2*3x + 1 + 32 = x – 8
x + 10 – 6x + 1 = x – 8 isolate the radical
10 + 8 = 6x + 1
18 = 6x + 1 div. by 6
3 = x + 1 square again
32 = (x + 1)2
Radical Equations

55. b. x + 1 – 3 = x – 8 square both sides;
(x + 1 – 3)2 = (x – 8)2
x + 1 – 2*3x + 1 + 32 = x – 8
x + 10 – 6x + 1 = x – 8 isolate the radical
10 + 8 = 6x + 1
18 = 6x + 1 div. by 6
3 = x + 1 square again
32 = (x + 1)2
9 = x + 1
8 = x This answer is good.
Radical Equations

56. Radical Equations
Exercise A. Isolate the radical then solve for x by squaring
both sides. Make sure to check your answers.
1. x = 3 2. x + 3 = 0 3. x – 5 = 3
5. 2x – 3 = 3
4. x – 5 = 3
6. 2x – 3 = 3 7. 2x – 3 = 3
8. 4x – 1 = 3 9. 4x – 1 = 3 10. 2x – 3 = – 3
11. 23x – 1 + 3 = 7 12. 4 – 33 – 2x = 1
13. x2 – 8 – 1 = 0 14. x2 – 8x – 3 = 0
Exercise B. Isolate one radical if needed, square. Then do it
again to solve for x. Make sure to check your answers.
15. x – 2 = x – 4 16. x + 3 = x + 1
17. 2x – 1 = x + 5 18. 4x + 1 – x + 2 = 1
19. x – 2 = x + 3 – 1 20. 3x + 4 = 3 – x – 1
21. 2x + 5 = x + 4 22. 5 – 4x – 3 – x = 1

57. Radical Equations
25. Given that (x, 4) is the distance of 5 from the origin
(0, 0). Find x and draw the points.
26. Find y where the points (2, y) is the distance of 5 from
(–1 , –1). Draw the points.
23. Given that (x, 0) has the same distance to (0, 2) as to
the point (2, –2). Find x and draw.
24. Given that (0, y) has the same distance to (3, 0) as to
the point (2, 1). Find x and draw.
Exercise C. Use the distance formula D = √Δx2 + Δ y2
to solve the following distance problems.