Question

1. What is the extraneous solution of √x−3 + x = 9?

2. If the volume of a cube is 108 ft^3, what is the side length? Keep your answer in radical form.

Can someone help with either of them? Thanks!

Answer 1

1. x = 12

2. 3∛4 feet.

Explanation:

1. We are given the equation of x as
`√(x - 3) + x = 9` and we have to find the extraneous solution of the equation.

Now,
`√(x - 3) + x = 9` ............. (1)

⇒
`√(x - 3) = 9 - x`

Squaring both sides we get,

⇒ x - 3 = 81 - 18x + x²

⇒ x² - 19x + 84 = 0

⇒ x² - 12x - 7x + 84 = 0

⇒ (x - 12)(x - 7) = 0

⇒ x = 12 or 7.

Now, putting x = 12 in the equation (1) we get,

`√(12 - 3) + 12 \\eq 9`

Again, putting x = 7 in the equation (1) we get,

`√(7 - 3) + 7 = 9`

Therefore, x = 12 is an extraneous solution of this equation. (Answer)

2. The volume of a cube with side lengths a ft. is given to be 108 ft³.

So, a³ = 108

⇒ a = 3∛4 feet.

Therefore, the length of the sides of the cube are each 3∛4 feet. (Answer)