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# Help ASAP

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54
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Let a and b be real numbers such that a - b = 4 and a-b3 = 52

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

Jul 2, 2023

#2
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$$a^3-b^3=(a-b)(a^2+ab+b^2)\\ sub \;\; a-b=4\qquad and \qquad a^3-b^3=52\\ 52=4(a^2+ab+b^2)\\ 13=(a^2+ab+b^2)\\ 13-3ab=a^2+ab+b^2-3ab\\ 13-3ab=a^2-2ab+b^2\\ 13-3ab=(a-b)^2\\ 13-3ab=4^2\\ -3ab=3\\ ab=-1\\~\\ 13=(a^2+ab+b^2)\\ 13+ab=(a^2+ab+b^2+ab)\\ 13+ab=(a^2+2ab+b^2)\\ 13+ab=(a+b)^2\\ 13-1=(a+b)^2\\ 12=(a+b)^2\\ a+b=\pm\sqrt{12}$$

So now check that i have not made any careless errors and solve simultaneously to get all possible values of a and b

Latex:

a^3-b^3=(a-b)(a^2+ab+b^2)\\
52=4(a^2+ab+b^2)\\
13=(a^2+ab+b^2)\\
13-3ab=a^2+ab+b^2-3ab\\
13-3ab=a^2-2ab+b^2\\
13-3ab=(a-b)^2\\
13-3ab=4^2\\
-3ab=3\\
ab=-1\\~\\
13=(a^2+ab+b^2)\\
13+ab=(a^2+ab+b^2+ab)\\
13+ab=(a^2+2ab+b^2)\\
13+ab=(a+b)^2\\
13-1=(a+b)^2\\
12=(a+b)^2\\
a+b=\pm\sqrt{12}

Jul 4, 2023
#3
+95
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