Question

(Many Points, Please Help!)

Solve the rational equation x/4 = x^2/x + 2 and check for extraneous solutions.


x = 2/3 ; 0 is an extraneous solution

x = 0; x = 2/3 is an extraneous solution

x = 0 and x = 2/3

No solution

Answer 1

x = 0 and x = 2/3 are solutions to the eqution

Explanation:

Hi there!

Let´s write the equation:

x/4 = x² / (x + 2)

Multiply both sides of the equation by (x + 2):

1/4 · x · (x + 2) = x²

Apply distributive property:

1/4 · x² + 1/2 · x = x²

Substract x² to both sides of the equation:

-x² + 1/4 · x² + 1/2 · x = 0

-3/4 · x² + 1/2 · x = 0

x(-3/4 · x + 1/2)

x = 0

and

-3/4 · x + 1/2 = 0

Substract 1/2 to both sides of the equation:

-3/4 · x = -1/2

divide by -3/4 both side of the equation:

x = -1/2 / -3/4

x = 2/3

Let´s check the solutions:

x = 0

0/4 = 0 / 0 + 2

0 = 0

x = 2/3

2/3 / 4 = (2/3)² / (2/3 + 2)

1/6 = 1/6

Then x = 0 and x = 2/3 are solutions to the eqution