+660 1. Let a and b be the solutions to \(7x^2+x-5=-4x-9\). Compute \((a-4)(b-4)\)
2. Let a and b be the roots of the quadratic \(2x^2-8x+7=0\). Compute \(a^3+b^3\).
keIhaku is not me btw, idk who they are or why they copied my name and pfp. I changed my pfp and also my username is Keihaku
+118670 1)
\(7x^2+x-5=-4x-9\\ 7x^2+5x+4=0\\\)
The sum of the roots a+b= -5/7
The product of the roots ab=4/7
\((a-4)(b-4)\\ =ab-4a-4b+16\\ =ab-4(a+b)+16\\~\\ \text{Substitute and you get}\\ =\frac{4}{7}-4*\frac{-5}{7}+16\\ =\frac{4}{7}+\frac{20}{7}+16\\ =\frac{24}{7}+16\\ =19\frac{3}{7}\)
+118670 the second one is very similar
You just have to know that
\(a^3+b^3=(a+b)(a^2-ab+b^2)\\ \text{rearranging I get}\\ a^3+b^3=(a+b)(a^2+2ab+b^2-3ab)\\ a^3+b^3=(a+b)([a+b]^2-3ab)\\ \text{now check what I have done and do the appropriate substitutions to finish the question.}\)
