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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 $b.$

 Feb 18, 2024
 #1
avatar+394 
+1

A)

\({(a-b)}^{3}= {a}^{3}-3{a}^{2}b+3a{b}^{2}-{b}^{3} = 1\)

\(1-3{a}^{2}b+3a{b}^{2}=1\)

\(3ab(b-a)=0\)

\(-3ab=0\)

\(ab=0\)

 Feb 18, 2024
 #2
avatar+394 
+1

B)

\({(a+b)}^2={(a-b)}^{2}+4ab\)

From previously, ab=0

\((a+b)^2={1}^{2}\)

\(a+b=\pm1\).

 Feb 18, 2024
 #3
avatar+394 
+1

C)

We know a-b = 1, and a+b = 1 or a+b = -1.

Consider two cases:

Case 1:

\(\begin{cases} a-b = 1\\ a+b=1 \end{cases}\)

\(2a = 2, a=1\)

\(\begin{cases} a=1 \\ b=0 \end{cases}\)

Case 2:

\(\begin{cases} a-b=1 \\ a+b = -1 \end{cases}\)

\(2a=0, a=0\)

\(\begin{cases} a=0 \\ b=-1 \end{cases}\)

Our two solutions are \(a, b = (1, 0)\) and \(a, b = (0, -1)\).

 Feb 18, 2024

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