+95 Let a and b be real numbers such that a - b = 4 and a3 -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
+118696
a3−b3=(a−b)(a2+ab+b2)suba−b=4anda3−b3=5252=4(a2+ab+b2)13=(a2+ab+b2)13−3ab=a2+ab+b2−3ab13−3ab=a2−2ab+b213−3ab=(a−b)213−3ab=42−3ab=3ab=−1 13=(a2+ab+b2)13+ab=(a2+ab+b2+ab)13+ab=(a2+2ab+b2)13+ab=(a+b)213−1=(a+b)212=(a+b)2a+b=±√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)\\
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}
