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. οx = 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. οx = 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. οx = 4 square each side
(οx)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. οx = 4 square each side
(οx)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. οx = 4 square each side
(οx)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. οx = 4 square each side
(οx)2 = 42
x = 16 It works.
b. οx β 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. οx = 4 square each side
(οx)2 = 42
x = 16 It works.
b. οx β 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. οx = 4 square each side
(οx)2 = 42
x = 16 It works.
b. οx β 3 = 4 square each side
(οx β 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. οx = 4 square each side
(οx)2 = 42
x = 16 It works.
b. οx β 3 = 4 square each side
(οx β 3)2 = 42
x β 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. οx = 4 square each side
(οx)2 = 42
x = 16 It works.
b. οx β 3 = 4 square each side
(οx β 3)2 = 42
x β 3 = 16
x = 19 It works.
18. c. ο2x + 1 = β3 square each side
(ο2x + 1)2 = (β3)2
Radical Equations
19. c. ο2x + 1 = β3 square each side
(ο2x + 1)2 = (β3)2
2x + 1 = 9
Radical Equations
20. c. ο2x + 1 = β3 square each side
(ο2x + 1)2 = (β3)2
2x + 1 = 9
2x = 8
Radical Equations
21. c. ο2x + 1 = β3 square each side
(ο2x + 1)2 = (β3)2
2x + 1 = 9
2x = 8
x = 4
Radical Equations
22. c. ο2x + 1 = β3 square each side
(ο2x + 1)2 = (β3)2
2x + 1 = 9
2x = 8
x = 4
However, x = 4 does not work. So there is no solution.
Radical Equations
23. c. ο2x + 1 = β3 square each side
(ο2x + 1)2 = (β3)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. ο2x + 1 = β3 square each side
(ο2x + 1)2 = (β3)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 Β± B)2 ο A2 Β± 2AB + B2
25. c. ο2x + 1 = β3 square each side
(ο2x + 1)2 = (β3)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. (οx + 4)2
For some problems, we have to square more than once to
eliminate all the radicals. Recall the squaring formula that
(A Β± B)2 ο A2 Β± 2AB + B2
26. c. ο2x + 1 = β3 square each side
(ο2x + 1)2 = (β3)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. (οx + 4)2
= (οx )2 + 2 * 4 οx + 42
For some problems, we have to square more than once to
eliminate all the radicals. Recall the squaring formula that
(A Β± B)2 ο A2 Β± 2AB + B2
27. c. ο2x + 1 = β3 square each side
(ο2x + 1)2 = (β3)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. (οx + 4)2
= (οx )2 + 2 * 4 οx + 42
= x + 8οx + 16
For some problems, we have to square more than once to
eliminate all the radicals. Recall the squaring formula that
(A Β± B)2 ο A2 Β± 2AB + B2
32. Radical Equations
b. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 4 )2
x + 2*4 οx + 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 4 )2
x + 2*4 οx + 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 4 )2
x + 2*4 οx + 16 = 5x + 4 isolate the radical
8οx = 4x β 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 4 )2
x + 2*4 οx + 16 = 5x + 4 isolate the radical
8οx = 4x β 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 4 )2
x + 2*4 οx + 16 = 5x + 4 isolate the radical
8οx = 4x β 12 divide by 4
2οx = x β 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 4 )2
x + 2*4 οx + 16 = 5x + 4 isolate the radical
8οx = 4x β 12 divide by 4
2οx = x β 3 square again
( 2οx)2 = (x β 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 4 )2
x + 2*4 οx + 16 = 5x + 4 isolate the radical
8οx = 4x β 12 divide by 4
2οx = x β 3 square again
( 2οx)2 = (x β 3)2
4x = x2 β 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 4 )2
x + 2*4 οx + 16 = 5x + 4 isolate the radical
8οx = 4x β 12 divide by 4
2οx = x β 3 square again
( 2οx)2 = (x β 3)2
4x = x2 β 2*3x + 9
0 = x2 β 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 4 )2
x + 2*4 οx + 16 = 5x + 4 isolate the radical
8οx = 4x β 12 divide by 4
2οx = x β 3 square again
( 2οx)2 = (x β 3)2
4x = x2 β 2*3x + 9
0 = x2 β 10x + 9 ο 0 = (x β 9)(x β 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 4 )2
x + 2*4 οx + 16 = 5x + 4 isolate the radical
8οx = 4x β 12 divide by 4
2οx = x β 3 square again
( 2οx)2 = (x β 3)2
4x = x2 β 2*3x + 9
0 = x2 β 10x + 9 ο 0 = (x β 9)(x β 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. (ο2x + 1 β 3)2
= (ο2x + 1)2 β 2*3ο2x + 1 + 32
= 2x + 1 β 6ο2x + 1 + 9
= 2x + 10 β 6ο2x+1
Example C. Solve for x.
a. οx + 4 = ο5x + 4 square both sides
(οx + 4)2 = (ο5x + 4 )2
x + 2*4 οx + 16 = 5x + 4 isolate the radical
8οx = 4x β 12 divide by 4
2οx = x β 3 square again
( 2οx)2 = (x β 3)2
4x = x2 β 2*3x + 9
0 = x2 β 10x + 9 ο 0 = (x β 9)(x β 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.
48. b. οx + 1 β 3 = οx β 8 square both sides;
(οx + 1 β 3)2 = (οx β 8)2
x + 1 β 2*3οx + 1 + 32 = x β 8
Radical Equations
49. b. οx + 1 β 3 = οx β 8 square both sides;
(οx + 1 β 3)2 = (οx β 8)2
x + 1 β 2*3οx + 1 + 32 = x β 8
x + 10 β 6οx + 1 = x β 8
Radical Equations
50. b. οx + 1 β 3 = οx β 8 square both sides;
(οx + 1 β 3)2 = (οx β 8)2
x + 1 β 2*3οx + 1 + 32 = x β 8
x + 10 β 6οx + 1 = x β 8 isolate the radical
Radical Equations
51. b. οx + 1 β 3 = οx β 8 square both sides;
(οx + 1 β 3)2 = (οx β 8)2
x + 1 β 2*3οx + 1 + 32 = x β 8
x + 10 β 6οx + 1 = x β 8 isolate the radical
10 + 8 = 6οx + 1
Radical Equations
52. b. οx + 1 β 3 = οx β 8 square both sides;
(οx + 1 β 3)2 = (οx β 8)2
x + 1 β 2*3οx + 1 + 32 = x β 8
x + 10 β 6οx + 1 = x β 8 isolate the radical
10 + 8 = 6οx + 1
18 = 6οx + 1
Radical Equations
53. b. οx + 1 β 3 = οx β 8 square both sides;
(οx + 1 β 3)2 = (οx β 8)2
x + 1 β 2*3οx + 1 + 32 = x β 8
x + 10 β 6οx + 1 = x β 8 isolate the radical
10 + 8 = 6οx + 1
18 = 6οx + 1 div. by 6
3 = οx + 1
Radical Equations
54. b. οx + 1 β 3 = οx β 8 square both sides;
(οx + 1 β 3)2 = (οx β 8)2
x + 1 β 2*3οx + 1 + 32 = x β 8
x + 10 β 6οx + 1 = x β 8 isolate the radical
10 + 8 = 6οx + 1
18 = 6οx + 1 div. by 6
3 = οx + 1 square again
32 = (οx + 1)2
Radical Equations
55. b. οx + 1 β 3 = οx β 8 square both sides;
(οx + 1 β 3)2 = (οx β 8)2
x + 1 β 2*3οx + 1 + 32 = x β 8
x + 10 β 6οx + 1 = x β 8 isolate the radical
10 + 8 = 6οx + 1
18 = 6οx + 1 div. by 6
3 = οx + 1 square again
32 = (οx + 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. οx = 3 2. οx + 3 = 0 3. οx β 5 = 3
5. 2οx β 3 = 3
4. οx β 5 = 3
6. 2οx β 3 = 3 7. ο2x β 3 = 3
8. ο4x β 1 = 3 9. ο4x β 1 = 3 10. ο2x β 3 = β 3
11. 2ο3x β 1 + 3 = 7 12. 4 β 3ο3 β 2x = 1
13. οx2 β 8 β 1 = 0 14. οx2 β 8x β 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. οx β 2 = οx β 4 16. οx + 3 = οx + 1
17. 2οx β 1 = οx + 5 18. ο4x + 1 β οx + 2 = 1
19. οx β 2 = οx + 3 β 1 20. ο3x + 4 = ο3 β x β 1
21. 2οx + 5 = οx + 4 22. ο5 β 4x β ο3 β 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
(β1 , β1). Draw the points.
23. Given that (x, 0) has the same distance to (0, 2) as to
the point (2, β2). 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 = βΞx2 + Ξ y2
to solve the following distance problems.