SOLUTION: The sum of the reciprocals of two numbers is 16/15 and the second number is 1 larger thank the first. Find the two numbers?

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Question 540359: The sum of the reciprocals of two numbers is 16/15 and the second number is 1 larger thank the first. Find the two numbers?
Answer by oberobic(2304) About Me  (Show Source):
You can put this solution on YOUR website!
The two numbers can be defined as:
x
x+1
.
so their reciprocals are:
1/x
1/(x+1)
.
1/x + 1/(x+1) = 16/15
.
((x+1)+x)/(x*(x+1) = 16/15
.
15((x+1)+x) = 16(x*(x+1))
.
15x + 15 + 15x = 16(x^2 +x)
.
30x + 15 = 16x^2 +16x
.
0 = 16x^2 -14x -15
.
16x^2 -14x -15 = 0
.
(16x+10)(x-3/2) = 0
.
Check with FOIL.
16x^2 -24x +10x -15
Correct
.
So, x = -5/8 or x=3/2
.
+graph%28500%2C500%2C-5%2C5%2C-20%2C20%2C16%2Ax%5E2-14%2Ax-15%29+
.
Or you could solve with quadratic equation.
.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 16x%5E2%2B-14x%2B-15+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-14%29%5E2-4%2A16%2A-15=1156.

Discriminant d=1156 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--14%2B-sqrt%28+1156+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-14%29%2Bsqrt%28+1156+%29%29%2F2%5C16+=+1.5
x%5B2%5D+=+%28-%28-14%29-sqrt%28+1156+%29%29%2F2%5C16+=+-0.625

Quadratic expression 16x%5E2%2B-14x%2B-15 can be factored:
16x%5E2%2B-14x%2B-15+=+16%28x-1.5%29%2A%28x--0.625%29
Again, the answer is: 1.5, -0.625. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+16%2Ax%5E2%2B-14%2Ax%2B-15+%29