SOLUTION: denominator of a fraction is 2 more than the numerator if the sum of fraction and its denominator is 28/5 find the fraction

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Question 1108819: denominator of a fraction is 2 more than the numerator if the sum of fraction and its denominator is 28/5 find the fraction
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
let x equal the numerator.

then x + 2 equal the denominator.

your equation becomes (x/(x+2) + (x+2) = 28/5.

multiply both sides of this equation by (x+2) and you get:

x + (x+2)^2 = 28/5 * (x+2).

multiply both sides of this equation by 5 to get:

5 * (x + (x+2)^2) = 28 * (x+2)

since (x+2)^2 x^2 + 4x + 4, this equation can be simplified to:

5 * (x + x^2 + 4x + 4) = 28 * (x + 2)

combine like terms to get:

5 * (x^2 + 5x + 4) = 28 * (x + 2)

since 28 * (x + 2) is equal to 28 * x + 56, this equation becomes:

5 * (x^2 + 5x + 4) = 28 * x + 56

since 5 * (x^2 + 5x + 4) = 5 * x^2 + 25 * x + 20, this equation becomes:

5 * x^2 + 25 * x + 20 = 28 * x + 56

subtract 28 * x and subtract 56 from both sides of this equation to get:

5 * x^2 - 3 * x - 36 = 0

factor this quadratic equation to get:

x = 3 or x = -2.4

replace x with 3 and then replace x with -2.4 to see which of these, if not both, are solutions to this problem.

when x = 3, the original equation of (x/(x+2)) + (x+2) = 28/5 becomes (3/5) + 5 = 5.6 which is equal to 28/5.

you can use your calculator to confirm that 28/5 = 5.6.

when x = -2.4, the original equation (x/(x+2)) + (x+2) = 28/5 becomes (-2.4/-.4) -2.4 + 2 which is equal to 5.6.

looks like both values of x satisfy the original equation, so the solution is that x can be 3 or x = -2.4.

you can also check by graphing the equations of:

y = (x/(x+2)) + (x+2) and y = 28/5, and then looking for the intersection of those 2 equations.

the intersections confirm the algebra.

the graph looks like this: