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# plz help

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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
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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
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Nevermind. Nothing here.

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