Exam

Olympiad workbook

Subject

IOQM

Chapter

Quadratic Equation

Olympiad workbook · IOQM · Quadratic Equation

Let a, b be integers such that all the roots of the equation (x^2+a...

Let $a, b$ be integers such that all the roots of the equation \left(x^2+a x+20\right)$\left(x^2+17 x+b\right)=0$ are negative integers. What is the smallest possible value of $\mathrm{a}+\mathrm{b}$ ?

Step-by-step solution

$$\begin{aligned} & \text { so } a>0 \text { and } b>0 \text { since sum of roots } < 0 \text { and product }>0 \\ & (\text { since } 20=(1 \times 20) \times(2 \times 10) \text { or }(4 \times 5) \\ & \min a=9 \\ & -17=\alpha+\beta \Rightarrow(\alpha \beta) \equiv(-1,-16),(-2,-15),(-8,-9) \\ & \min b=16 \\ & (a+b)_{\min }=a_{\min }+b_{\min }=9+16=25\end{aligned}$$