Question

The number obtained by adding the squares of a number and its reciprocal is greater by 7/4 then the sum of the number and its reciprocal. Find the number.

Answer 1

Step-by-step explanation:

find x such that

x%5E2%2B%281%2Fx%29%5E2+=+x%2B1%2Fx%2B7%2F4

Let us say that x%2B1%2Fx=y .

Then, y%5E2=x%5E2%2B%281%2Fx%29%5E2%2B2x%281%2Fx%29 ,

y%5E2=x%5E2%2B%281%2Fx%29%5E2%2B2 , and

x%5E2%2B%281%2Fx%29%5E2=y%5E2-2 .

With that, we can re-write the equation as

y%5E2-2=y%2B7%2F4 .

Multiplying both sides of the equal sign times 4 , we get

4y%5E2-8=4y%2B7 ,

which simplifies to

4y%5E2-4y-15=0

Either applying the quadratic formula, or factoring,

or "completing the square", we can find the solutions to be

y=5%2F2 and y=-3%2F2 .

x%2B1%2Fx=-3%2F2 does not have real solutions,

but we can find real solutions for

x%2B1%2Fx=5%2F2 .

Multiplying both sides of the equal sign times 2x ,

we get the equivalent equation

2x%5E2%2B2x=5<--->2x%5E2%2B2x-5=0 ,

whose solutions are

highlight%28x=2%29 and (of course) highlight%28x=1%2F2%29 .