# SOLUTION: Could somebody guide me about this which is about the relation between the roots and the coefficients? For what values of m will the equation {{{x^2-4x+7=m(x-1)}}} have a) one ro

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 Click here to see ALL problems on Quadratic Equations Question 202022: Could somebody guide me about this which is about the relation between the roots and the coefficients? For what values of m will the equation have a) one root the reciprocal of the other, b) one root equal zero c) equal roots Answer by Edwin McCravy(8912)   (Show Source): You can put this solution on YOUR website!Could somebody guide me about this which is about the relation between the roots and the coefficients? For what values of m will the equation have a) one root the reciprocal of the other, b) one root equal zero c) equal roots ``` To do any of those we have to first get in the form ---------------- ``` a) one root the reciprocal of the other ``` Suppose one root is r and the other is , then the quadratic equation with this property and leading coefficient 1 is found this way: Therefore this must be the same equation as So we equate like parts: or simplifying: To check that, we substitute for : , , So both roots are equal and -1, but -1 is the reciprocal of -1. So the answer to (a) is ------------- ``` b) one root equal zero ``` Suppose one root is 0 and the other is r, then the quadratic equation with this property and leading coefficient 1 is found this way: Therefore this must be the same equation as So we equate like parts: Simplifying: So we see that m must be -7. To check that, we substitute for : , One root is 0, so we are correct. So the answer to (b) is ------------- ``` c) equal roots ``` Suppose one root is r and the other is also r, then the quadratic equation with this property and leading coefficient 1 is found this way: Therefore this must be the same equation as So we equate like parts: Simplifying: Solve both equations for m Equate the right sides since both equal m: , , Substituting into Substituting into We don't need to check m=-6 for that was the value of m in part (a), and we knew that in that case the roots were not only reciprocals but they were also equal. Checking m=2 , , So there are two answers to (c), and . Edwin```