AMC 12 Mock Exam 5 Problem 9

Problem 9 of 25

Real numbers $b$ and $c$ have the following property: the roots of $x^2+bx+c$ are $r$ and $s$ , and the roots of $x^2+cx+b$ are $r+1$ and $s+1$ . What is $b+c$ ?

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Solution

By Vieta's formulas, $r+s=-b$ and $rs=c$ , and also $(r+1)+(s+1)=-c$ and $(r+1)(s+1)=b$ . The third equation gives $-b+2=-c$ , i.e., $c=b-2$ . The fourth gives $rs+(r+s)+1=b$ , i.e., $c-b+1=b$ , so $c=2b-1$ . Hence $b-2=2b-1$ , giving $b=-1$ and $c=-3$ , and $b+c=\boxed{-4}$ .

(Check: $x^2-x-3$ has roots $\frac{1\pm\sqrt{13}}{2}$ , and adding $1$ gives $\frac{3\pm\sqrt{13}}{2}$ , the roots of $x^2-3x-1$ .)

Tagged: quadratics, Vieta's formulas

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