Question

Solve the equations in Parts A and B using inverse operations. Check your solutions. In your final answer, include all of your work. Part A: 5 + x2 = 2x2 + 13 Part B: 5 + x3 = 2x3 + 13

Answer 1

Part A:

To solve the equation 2x^2 + 5 = 13 using inverse operations, we need to isolate the variable x.

1. Subtract 5 from both sides of the equation to get rid of the constant term:

2x^2 + 5 - 5 = 13 - 5

2x^2 = 8

2. Divide both sides of the equation by 2 to solve for x:

(2x^2) / 2 = 8 / 2

x^2 = 4

3. Take the square root of both sides of the equation to solve for x:

√(x^2) = √4

x = ±2

Therefore, the solutions for Part A are x = 2 and x = -2.

To check these solutions, substitute them back into the original equation:

When x = 2:

2(2)^2 + 5 = 13

8 + 5 = 13

13 = 13 (true)

When x = -2:

2(-2)^2 + 5 = 13

8 + 5 = 13

13 = 13 (true)

Both solutions satisfy the original equation, so they are correct.

Part B:

To solve the equation 2x^3 - 13 = 5 using inverse operations, we need to isolate the variable x.

1. Add 13 to both sides of the equation to get rid of the constant term:

2x^3 - 13 + 13 = 5 + 13

2x^3 = 18

2. Divide both sides of the equation by 2 to solve for x:

(2x^3) / 2 = 18 / 2

x^3 = 9

3. Take the cube root of both sides of the equation to solve for x:

∛(x^3) = ∛9

x = ∛9

Therefore, the solution for Part B is x = ∛9.

To check this solution, substitute it back into the original equation:

When x = ∛9:

2(∛9)^3 - 13 = 5

2(9) - 13 = 5

18 - 13 = 5

5 = 5 (true)

The solution satisfies the original equation, so it is correct.

Answer 2

A).
`x=√(8i)`

B).
`x=2i`

Explanation:

Part A

`5+x^(2) =2*x^(2) +13\\5-5+x^(2)-2*x^(2) =2*x^(2) -2*x^(2)+13-5 \\x^(2)-2*x^(2)=13-5\\-x^(2)=8\\x^(2)=-8\\x=√(8i)`

Check:

`5+(√(8i))^(2)=2*(√(8i))^(2)+13`

`5+8i^(2)=2*8i^(2)+13\\5+8(-1)=2*8(-1)+13\\5-8=-16+13\\-3=-3`

Part B

`5+x^(3)=2*x^(3)+13\\5-5+x^(3)=2*x^(3)-2*x^(3)+13-5\\x^(3)-2*x^(3)=13-5\\-x^(3)=8\\x^(3)=-8\\x=-8^{(1)/(3) } \\x=8i^{(1)/(3) }\\x=2i`

Check:

`5+(2i)^(3)=2*(2i)^(3)+13\\5+8i^(3)=2*8i^(3)+13\\5+8-i=16-i+13\\5-8i=-16i+13\\5-13-8i+8i=-16i+8i+13-13\\-8=-8i\\-8=-8`