Question

Find all solutions to


`\sqrt[3]{15x-1} + \sqrt[3]{13x+1} =4\sqrt[3]{x}`
Enter all the separated, seperated by commas.

Answer 1

x= 0 ,
`(1)/(14)` ,
`(-1)/(12)`

Explanation:

Given, equation is
`\sqrt[3]{15x-1}` +
`\sqrt[3]{13x+1}` =
`4\sqrt[3]{x}`. →→→ (1)

Now, by cubing the equation on both sides, we get

(
`\sqrt[3]{15x-1}` +
`\sqrt[3]{13x+1}` )³ = (
`4\sqrt[3]{x}`)³

⇒ (15x-1) + (13x+1) + 3×
`\sqrt[3]{15x-1}`×
`\sqrt[3]{13x+1}` (
`\sqrt[3]{15x-1}` +
`\sqrt[3]{13x+1}`) = 64 x.

⇒ 28x + 3×
`\sqrt[3]{15x-1}`×
`\sqrt[3]{13x+1}` (
`4\sqrt[3]{x}`) = 64x.

(since from (1),
`\sqrt[3]{15x-1}` +
`\sqrt[3]{13x+1}` =
`4\sqrt[3]{x}`)

⇒ 12×
`\sqrt[3]{15x-1}`×
`\sqrt[3]{13x+1}` (
`\sqrt[3]{x}`)= 36x.

⇒ 3x =
`\sqrt[3]{(15x-1)(13+1)(x)}`.

Now, once again cubing on both sides, we get

(3x)³ = (
`\sqrt[3]{(15x-1)(13+1)(x)}`)³.

⇒ 27x³ = (15x-1)(13x+1)(x).

⇒ 27x³ = 195x³ + 2x² - x

⇒ 168x³ + 2x² - x = 0

⇒ x(168x² + 2x -1) = 0

⇒ by, solving the equation we get ,

x = 0 ; x =
`(1)/(14)` ; x =
`(-1)/(12)`

therefore, solution is x= 0 ,
`(1)/(14)` ,
`(-1)/(12)`