The reciprocal of a number minus the reciprocal of two less than the number is -2/15. find the number.
1/x - 1/(x-2) = -2/15
Two less than the reciprocal of a number x
To find the number, let's break down the given information step by step:
Let's assume the number is "x." According to the given information, we can express the following equation:
1/x - 1/(x-2) = -2/15
Now, to solve for x, we will first get rid of the fractions by multiplying through by the least common denominator (LCD), which in this case is 15x(x-2):
15(x-2) - 15x = -2x(x-2)
Now, let's simplify and expand the equation:
15x - 30 - 15x = -2x^2 + 4x
Combine like terms:
-30 = -2x^2 + 4x
Rearrange the equation to set it equal to zero:
2x^2 - 4x - 30 = 0
Next, let's simplify the equation by dividing through by 2:
x^2 - 2x - 15 = 0
Now, we need to factor the quadratic equation:
(x - 5)(x + 3) = 0
Using the zero product property, we can set each factor equal to zero:
x - 5 = 0 or x + 3 = 0
Solving for x, we get two possible solutions:
x = 5 or x = -3
Therefore, the number can either be 5 or -3.