find the positive value of x, to the nearest hundredth, that satisfies the equation 1 over x= x+2 over 2x+3

To find the positive value of x that satisfies the equation (1/x) = (x + 2)/(2x + 3), we need to solve for x.

First, let's clear the denominators by multiplying both sides of the equation by x(2x + 3):

x(2x + 3) * (1/x) = x(2x + 3) * (x + 2)/(2x + 3)

Simplifying this expression, we get:

2x + 3 = (x + 2)x

Now, distribute on the right side of the equation:

2x + 3 = x^2 + 2x

Rearranging the equation to set it equal to zero:

x^2 - 2x - 2x - 3 = 0

x^2 - 4x - 3 = 0

Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, factoring is not straightforward and completing the square may involve decimals, so let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -4, and c = -3. Substituting these values into the quadratic formula, we get:

x = (-(-4) ± √((-4)^2 - 4(1)(-3))) / (2(1))

Simplifying further:

x = (4 ± √(16 + 12)) / 2

x = (4 ± √28) / 2

x = (4 ± 2√7) / 2

Simplifying:

x = 2 ± √7

Since we are looking for the positive value to the nearest hundredth, we can substitute the plus sign:

x ≈ 2 + √7

Evaluating this expression to the nearest hundredth:

x ≈ 2 + 2.65

x ≈ 4.65

Therefore, the positive value of x, to the nearest hundredth, that satisfies the equation (1/x) = (x + 2)/(2x + 3) is approximately 4.65.