Question

Solve the rational equation x divided by 4 equals x squared divided by quantity x plus 2 end quantity, and check for extraneous solutions.

Answer 1

x = 0, x = 2/3

Explanation:

x / 4 = x² / (x + 2)

x = 4x² / (x + 2)

x * (x + 2) = 4x²

x² + 2x = 4x²

3x² - 2x = 0

x(3x - 2) = 0

x = 0, x = 2/3

Answer 2

x=0, 2/3

Explanation:

x/4 = x^2 /(x+2)

Using cross products

4x^2 = x(x+2)

Distribute

4x^2 = x^2 +2x

Subtract x^2 from each side

3x^2 = 2x

Subtract 2x from each side

3x^2 -2x =0

Factor out x

x(3x-2) =0

Using the zero product property

x=0 3x-2 =0

x=0 3x=2

x=0 x= 2/3

Check

x=0

0/4 = 0^2 /(0+2)

0 = 0

Check

x=2/3

(2/3)/4 = (2/3)^2 /((2/3)+2)

1/6 = (4/9)/ (8/3)

1/6 = 4/9 * 3/8

1/6 = 4/8 *3/9

1/6 = 1/6

Both solutions are valid