a fraction is such that the numerator is seven less than the square of the denominator. if six is added to the numerator and on to the denominator the result is 2. Find the fraction
a fraction is such that the numerator is seven less than the square of the denominator. if six is added to the numerator and on to the denominator the result is 2. Find the fraction
n = d^2-7
(n+6)/(d+6) = 2
I suspect a typo. wolframalpha gets
8+2√14
-----------
1+√14
However, if 1 is added to the denominator, then the fraction is
2/3
http://www.wolframalpha.com/input/?i=solve+n+%3D+d^2-7%2C+%28n%2B6%29%2F%28d%2B6%29+%3D+2
To solve this problem, let's try to set up an equation based on the given information.
Let's assume that the denominator of the fraction is represented by "x". Therefore, the numerator can be represented as "x^2 - 7" because it is seven less than the square of the denominator.
Now, we are told that if we add six to both the numerator and denominator, the result is 2. This can be written as:
(x^2 - 7 + 6) / (x + 6) = 2
Simplifying this equation:
(x^2 - 1) / (x + 6) = 2
Now we can cross-multiply:
x^2 - 1 = 2(x + 6)
Expanding the right side:
x^2 - 1 = 2x + 12
Rearranging the equation to bring all terms to one side:
x^2 - 2x - 13 = 0
At this point, we can solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's solve it using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the equation form: ax^2 + bx + c = 0, our values are:
a = 1, b = -2, and c = -13.
Substituting these values into the quadratic formula:
x = (-(-2) ± √((-2)^2 - 4(1)(-13))) / (2(1))
Simplifying:
x = (2 ± √(4 + 52)) / 2
x = (2 ± √56) / 2
x = 1 ± √14
Since it is not possible to have a negative value for the denominator in a fraction, we will choose the positive value:
x = 1 + √14
Now we have the value of the denominator. To find the numerator, we use the expression "x^2 - 7":
Numerator = (1 + √14)^2 - 7
= 1 + 2√14 + 14 - 7
= 8 + 2√14
Therefore, the fraction is (8 + 2√14) / (1 + √14).