The sum of a number and its reciprocal is 37/6. Find the number.
x + 1/x = 37/6 = 6 + 1/6
76
To find the number, let's assume that the number is x.
According to the given information, the sum of a number (x) and its reciprocal (1/x) is equal to 37/6. This can be written as an equation:
x + (1/x) = 37/6
To solve this equation, we can multiply the entire equation by 6x to eliminate the denominators:
6x(x) + 6x(1/x) = (37/6)(6x)
Simplifying this equation gives us:
6x^2 + 6 = 37x
Now, we have a quadratic equation. Let's rearrange it to bring all terms to one side:
6x^2 - 37x + 6 = 0
Next, we can either factor the quadratic equation or use the quadratic formula to find the values of x. Since factoring might not be convenient in this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 6, b = -37, and c = 6. Substituting these values into the quadratic formula gives us:
x = (-(-37) ± √((-37)^2 - 4(6)(6))) / (2(6))
Simplifying this equation gives us:
x = (37 ± √(1369 - 144)) / 12
x = (37 ± √(1225)) / 12
Now, let's calculate the two possible values for x:
x₁ = (37 + √1225) / 12
x₁ = (37 + 35) / 12
x₁ = 72 / 12
x₁ = 6
x₂ = (37 - √1225) / 12
x₂ = (37 - 35) / 12
x₂ = 2 / 12
x₂ = 1/6
Therefore, the two possible numbers are 6 and 1/6.
sum=N+1/N=37/6
N^2-37N/6+1=0
N=(37/6 +-sqrt( (37/6)^2 -4*1*1) /2=37/12 +- sqrt (34/4)
N=6
check: 6+1/6=37/6