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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 

 Mar 11, 2024
 #4
avatar+118617 
+4

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

 

\((a-4)(b-4)\\ =ab-4a-4b+16\\ =ab-4(a+b)+16\\~\\ \text{Substitute and you get}\\ =\frac{4}{7}-4*\frac{-5}{7}+16\\ =\frac{4}{7}+\frac{20}{7}+16\\ =\frac{24}{7}+16\\ =19\frac{3}{7}\)

 Mar 17, 2024
 #5
avatar+118617 
+4

the second one is very similar 

You just have to know that 

 

     \(a^3+b^3=(a+b)(a^2-ab+b^2)\\ \text{rearranging I get}\\ a^3+b^3=(a+b)(a^2+2ab+b^2-3ab)\\ a^3+b^3=(a+b)([a+b]^2-3ab)\\ \text{now check what I have done and do the appropriate substitutions to finish the question.}\)

 Mar 17, 2024

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