A sum of a number and it's reciprocal is 2 1/30 (mixed fraction) determine the number?
I have tried solving it with no luck. I get stuck on the part of completing the square. Please help me
Thank you
n + 1/n = 61/30
n^2 + 1 = 61/30 n
n^2 - 61/30 n + 1 = 0
use the quadratic formula to find n
To solve this problem, we can start by assigning a variable to the unknown number. Let's say the unknown number is represented by the variable "x".
According to the problem, the sum of a number (x) and its reciprocal (1/x) is equal to 2 1/30.
To represent 2 1/30 as an improper fraction, we convert the whole number part to a fraction by multiplying it by the denominator (30) and adding the numerator (1). This gives us the fraction 61/30.
So, the equation to solve is: x + (1/x) = 61/30.
To eliminate the fraction, we can multiply both sides of the equation by 30x, the least common multiple of the denominators.
30x(x) + 30x(1/x) = 30x(61/30).
This simplifies to: 30x^2 + 30 = 61x.
Next, we move all the terms to one side to form a quadratic equation:
30x^2 - 61x + 30 = 0.
The next step requires factoring or applying the quadratic formula to solve for x. Since you mentioned getting stuck at the step of completing the square, let's use the quadratic formula instead.
The quadratic formula states that for an equation in the form of ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a).
For our equation, a = 30, b = -61, and c = 30. Plugging these values into the quadratic formula, we get:
x = (-(-61) ± sqrt((-61)^2 - 4(30)(30))) / (2(30)).
Simplifying this expression:
x = (61 ± sqrt(3721 - 3600)) / 60.
x = (61 ± sqrt(121)) / 60.
The square root of 121 is 11, so we can simplify further:
x = (61 ± 11) / 60.
This gives us two possible solutions for x:
x₁ = (61 + 11) / 60 = 72/60 = 6/5.
x₂ = (61 - 11) / 60 = 50/60 = 5/6.
So, the possible values for the number x are 6/5 and 5/6.