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The real numbers $a$ and $b$ satisfy $a - b = 1$ and $a^3 - b^3 = 1.$

(a) Find all possible values of $ab.$
(b) Find all possible values of $a + b.$
(c) Find all possible values of $a$ and $b.$

 Nov 11, 2023
 #1
avatar+22 
0

a - b = 1

a = b + 1

a^3 = (b+1)^3

= b^3 +3b^2 +3b + 1

 

As mentioned in the question above, a^3 - b^3 = 1

We have proven a^3 = b^3 +3b^2 +3b + 1

So, replacing a^3 with b^3 +3b^2 +3b + 1:

 

b^3 +3b^2 +3b + 1 - b^3 = 1

 

Collecting like terms

3b^2 + 3b = 0

3b^2 = -3b

b^2 = -b

b = -1, b = 0

 

If b = -1:

a-(-1) = 1

a + 1 = 1

a = 0

 

If b = 0:

a - 0 = 1

a = 1

 

So if b = -1, a = 0

If b = 0, a = 1

 

Part (a)

It can be either -1 * 0

Which equals 0

Or 0 * 1

Which equals 0

So the only answer is 0

 

Part (b)

It can be either -1 + 0

Which equals -1

Or 0 + 1

Which equals 1

So the answers are 1, -1

 

Part (c)

The possible values for a are 0, 1

The possible values for b are -1, 0

 Nov 12, 2023
 #2
avatar+22 
0

Nevermind. Nothing here.

Darkness  Nov 12, 2023
edited by Darkness  Nov 12, 2023

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