Thread: A Generalized Problem Problem

Hey all,

So in the shower, I found myself trying to generalize a problem. I was discussing with my students earlier how the two numbers, x and y, that both add and multiply to 4
are both 2. In short, 2+2 and 2*2 are both 4. I was trying to generalize this pattern to work for numbers other than 4. Here is the system I am starting with.

x + y = A
xy = A

Step 1)
x = A-y, so (A-y)y = A, so Ay - y^2 = A, so -y^2 + Ay - A = 0 (quadratic formula time..)

a= -1
b = A
c = -A

{-A + sqrt[A^2 -4(-1)(-A)] } / [2(-1)]

Simplified..

{-A + sqrt(A^2 - 4A)} / -2

Simplified..

A/2 + sqrt(A^2 - 4A)

To check, I plugged in A = 4 (because then I know my answer should be 2)

4/2 + sqrt(16-16) = 2

To see what the problem would be with A = 6..

6/2 + sqrt(36 - 24) = 3 + sqrt(12)

So, I plugged this into my original system to solve for the other variable.

[3 + sqrt(12)] + y = 6, so y must be 6 - 3 - sqrt(12) = 3 - sqrt(12).

So, x + y = 3+sqrt(12) + 3 - sqrt (12), which is 6. First equation checks.

Second equation is what is giving me a headache...

xy = 6

(3 + sqrt(12))*((3 - sqrt(12)) = 6

9 - 3sqrt(12) + 3sqrt(12) - 12 = 6

9 - 12 = 6 (does not work)

Which error am I making?

Cheers,
Ted

Hello.

I have made a comment inside of your work:

Originally Posted by rdtedm

Hey all,

So in the shower, I found myself trying to generalize a problem. I was discussing with my students earlier how the two numbers, x and y, that both add and multiply to 4
are both 2. In short, 2+2 and 2*2 are both 4. I was trying to generalize this pattern to work for numbers other than 4. Here is the system I am starting with.

x + y = A
xy = A

Step 1)
x = A-y, so (A-y)y = A, so Ay - y^2 = A, so -y^2 + Ay - A = 0 (quadratic formula time..)

a= -1
b = A
c = -A

{-A + sqrt[A^2 -4(-1)(-A)] } / [2(-1)]

Simplified..

{-A + sqrt(A^2 - 4A)} / -2

Simplified..

A/2 + sqrt(A^2 - 4A) It looks like you forgot to divide the 2 into the term with the square root, so it should have been A/2 + sqrt(A^2-4A)/(-2).

To check, I plugged in A = 4 (because then I know my answer should be 2)

4/2 + sqrt(16-16) = 2

To see what the problem would be with A = 6..

6/2 + sqrt(36 - 24) = 3 + sqrt(12)

So, I plugged this into my original system to solve for the other variable.

[3 + sqrt(12)] + y = 6, so y must be 6 - 3 - sqrt(12) = 3 - sqrt(12).

So, x + y = 3+sqrt(12) + 3 - sqrt (12), which is 6. First equation checks.

Second equation is what is giving me a headache...

xy = 6

(3 + sqrt(12))*((3 - sqrt(12)) = 6

9 - 3sqrt(12) + 3sqrt(12) - 12 = 6

9 - 12 = 6 (does not work)

Which error am I making?

Cheers,
Ted

I realized my mistake on my way to lunch. Thank you!

Ted