Let f(x)=x^2+a x+b . If for all nonzero real x f (x+(1)/(x))=f(x)+f ((1)/(x)) and the roots of f(x)=0 are integers, what is the value of a^2+b^2 ?

Correct answer: 13 aligned & f(x+(1)/(x))=f(x)+f((1)/(x)) & (x+(1)/(x))^2+a(x+(1)/(x))+b=x^2+a x+b+(1)/(x^2)+(a)/(x)+b & b=2 & α β=2 (α, β)=(1,2) or (-1,-2) & a=3 or -3 a^2+b^2=9+4=13 aligned

Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Quadratic Equation

Olympiad workbook · IOQM · Quadratic Equation

Let f(x)=x^2+a x+b . If for all nonzero real x f (x+(1)/(x))=f(x)+f...

Let

f(x)=x^2+a x+b

. If for all nonzero real

x

f

\left(x+\frac{1}{x}\right)=f(x)+f

\left(\frac{1}{x}\right)

and the roots of

f(x)=0

are integers, what is the value of

a^2+b^2 ?

Correct answer

Step-by-step solution

\begin{aligned} & f\left(x+\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right) \\ & \Rightarrow\left(x+\frac{1}{x}\right)^2+a\left(x+\frac{1}{x}\right)+b=x^2+a x+b+\frac{1}{x^2}+\frac{a}{x}+b \\ & \Rightarrow b=2 \\ & \Rightarrow \alpha \beta=2 \quad \Rightarrow(\alpha, \beta)=(1,2) \text { or }(-1,-2) \\ & \Rightarrow a=3 \text { or }-3 \quad \Rightarrow a^2+b^2=9+4=13\end{aligned}

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