Radical Equations

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

Radical Equations

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

Radical Equations

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

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.

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.

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

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

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

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

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.

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

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

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

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

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.

c. 2x + 1 = –3Radical Equations

c. 2x + 1 = –3 square each side (2x + 1)2 = (–3)2

Radical Equations

c. 2x + 1 = –3 square each side (2x + 1)2 = (–3)2

2x + 1 = 9

Radical Equations

c. 2x + 1 = –3 square each side (2x + 1)2 = (–3)2

2x + 1 = 92x = 8

Radical Equations

c. 2x + 1 = –3 square each side (2x + 1)2 = (–3)2

2x + 1 = 92x = 8 x = 4

Radical Equations

c. 2x + 1 = –3 square each side (2x + 1)2 = (–3)2

2x + 1 = 92x = 8 x = 4

However, x = 4 does not work. So there is no solution.

Radical Equations

c. 2x + 1 = –3 square each side (2x + 1)2 = (–3)2

2x + 1 = 92x = 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.

c. 2x + 1 = –3 square each side (2x + 1)2 = (–3)2

2x + 1 = 92x = 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

c. 2x + 1 = –3 square each side (2x + 1)2 = (–3)2

2x + 1 = 92x = 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.

c. 2x + 1 = –3 square each side (2x + 1)2 = (–3)2

2x + 1 = 92x = 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.

c. 2x + 1 = –3 square each side (2x + 1)2 = (–3)2

2x + 1 = 92x = 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 + 8x + 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.

Radical Equationsb. (2x + 1 – 3)2

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+1

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+1 When squaring both sides of an equation to remove a radical, make sure that the radical term is isolated to one side first.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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 8x = 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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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 8x = 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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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 8x = 4x – 12 divide by 4 2x = 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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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 8x = 4x – 12 divide by 4 2x = x – 3 square again ( 2x)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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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 8x = 4x – 12 divide by 4 2x = x – 3 square again ( 2x)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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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 8x = 4x – 12 divide by 4 2x = x – 3 square again ( 2x)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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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 8x = 4x – 12 divide by 4 2x = x – 3 square again ( 2x)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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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 8x = 4x – 12 divide by 4 2x = x – 3 square again ( 2x)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.

Radical Equationsb. (2x + 1 – 3)2 = (2x + 1)2 – 2*32x + 1 + 32

= 2x + 1 – 62x + 1 + 9 = 2x + 10 – 62x+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 8x = 4x – 12 divide by 4 2x = x – 3 square again ( 2x)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.

b. x + 1 – 3 = x – 8Radical Equations

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

Radical Equations

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

Radical Equations

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

Radical Equations

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

Radical Equations

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

Radical Equations

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

Radical Equations

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

Radical Equations

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

Radical Equations

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

9 = x + 1 8 = x This answer is good.

Radical Equations

Radical EquationsExercise 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 = 35. 2x – 3 = 3

4. x – 5 = 3 6. 2x – 3 = 3 7. 2x – 3 = 3

8. 4x – 1 = 3 9. 4x – 1 = 3 10. 2x – 3 = – 311. 23x – 1 + 3 = 7 12. 4 – 33 – 2x = 113. x2 – 8 – 1 = 0 14. x2 – 8x – 3 = 0Exercise 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 + 117. 2x – 1 = x + 5 18. 4x + 1 – x + 2 = 119. x – 2 = x + 3 – 1 20. 3x + 4 = 3 – x – 121. 2x + 5 = x + 4 22. 5 – 4x – 3 – x = 1

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.