Product and sum of two numbers

[Stochastic13]

Homework Statement

If the product of two numbers if 5, and the sum of these same two numbers is 4, what's the sum of the reciprocals of these two numbers?

Homework Equations

The Attempt at a Solution

I tried:

let's say we have two numbers a and b, then

a*b = 5
a = 5/b

since their product is 4, we get

b + 5/b = 4
or
b^2 -4b + 5 = 0

but this has no real solutions just imaginary
because this function is > 0 for all b

I can't factor it and the quadratic formula doesn't work
since 4^2 < 4*1*5

also the derivative is 2b -4 = 0
so b = 2 is the minimum since second derivative is positive
but that doesn't help either.

What do I do?

 

[chiro]

Homework Statement

If the product of two numbers if 5, and the sum of these same two numbers is 4, what's the sum of the reciprocals of these two numbers?

Homework Equations

The Attempt at a Solution

I tried:

let's say we have two numbers a and b, then

a*b = 5
a = 5/b

since their product is 4, we get

b + 5/b = 4
or
b^2 -4b + 5 = 0

but this has no real solutions just imaginary
because this function is > 0 for all b

I can't factor it and the quadratic formula doesn't work
since 4^2 < 4*1*5

also the derivative is 2b -4 = 0
so b = 2 is the minimum since second derivative is positive
but that doesn't help either.

What do I do?

Hint: 1/a + 1/b = (a+b)/(ab)

 

[Stochastic13]

Using (a+b)/(ab) = 5
I get b = a/(5a - 1) and then I'm stuck...

 

[chiro]

Using (a+b)/(ab) = 5
I get b = a/(5a - 1) and then I'm stuck...

You were asked to get the sum of the reciprocals of a and b, or in other words calculate 1/a + 1/b. So based on this (and the hint), what do you think 1/a + 1/b is?

 

[Hurkyl]

the quadratic formula doesn't work
since 4^2 < 4*1*5

Why do you think that means the quadratic formula doesn't work?

 

[icystrike]

Sum of 2 reciprocals is sum of the numbers divided by product of numbers

 

[Char. Limit]

Using (a+b)/(ab) = 5

I don't see where you got that. You know that ab=5 and a+b=4, though. And you know that (a+b)/(ab) = a^-1 + b^-1. That's all you need.

 

[Stochastic13]

OK, so the answer is 4/5. I feel stupid for not seeing the answer myself.
To answer the question why quad formula doesn't work is that since 16 < 20 the square root of these numbers gives you a complex number.
Thanks again all.

 

[Hurkyl]

To answer the question why quad formula doesn't work is that since 16 < 20 the square root of these numbers gives you a complex number.

So? You already knew the roots were complex...