If a number is added to its reciprocal, the sum is 13/6, find the number.
x + 1/x = 13/6
x is 2/3 or 3/2
What are the steps?
To find the number, let's assume that the number is represented by 'x'. According to the given information, when the number is added to its reciprocal, the sum is 13/6.
Mathematically, we can express this as:
x + 1/x = 13/6
To solve this equation and find the value of 'x', we can multiply both sides by 6*x (to eliminate the denominator):
6*x * (x + 1/x) = 6*x * (13/6)
Simplifying this equation gives:
6*x^2 + 6 = 13*x
Now, let's rearrange the equation to put it in quadratic form:
6*x^2 - 13*x + 6 = 0
To solve this quadratic equation, we can factorize it or use the quadratic formula. In this case, factoring is simpler:
(2*x - 3)(3*x - 2) = 0
Setting each factor equal to zero gives two possible solutions:
2*x - 3 = 0, or 3*x - 2 = 0
Solving these equations for 'x' gives:
2*x = 3, or 3*x = 2
x = 3/2, or x = 2/3
Hence, the two possible values for 'x' are 3/2 and 2/3.