Exam

Olympiad workbook

Subject

IOQM

Chapter

Functions

Olympiad workbook · IOQM · Functions

The sum of [x] for all real numbers x satisfying the equation 16+15...

The sum of $[x]$ for all real numbers $x$ satisfying the equation $16+15 x+15 x^2=[x]^3$ is:

Step-by-step solution

$$\begin{aligned} & {[x]^3=15 x^2+15 x+16} \\ & {[x] \in(x-1, x]} \\ & {[x]^3 \in\left((x-1)^3, x^3\right]} \\ & \Rightarrow \quad 15 x^2+15 x+16 \in\left((x-1)^3, x^3\right] \\ & \quad x^3 \geq 15 x^2+15 x+16 \\ & \Rightarrow \quad x^3-15 x^2-15 x-16 \geq 0 \\ & \quad(x-16)\left(x^2+x+1\right) \geq 0 \\ & \Rightarrow \quad x \geq 16 \end{aligned}$$ Also, $15 x^2+15 x+16>x^3-3 x^2+3 x-1$ $\Rightarrow \quad x^3-18 x^2-12 x-17 < 0$ Let $f(x)=x^3-18 x^2-12 x-17$ $f(x)=3 x^2-36 x-12$
$$\begin{aligned} & f(\alpha) < 0 \\ & \Rightarrow \quad f(x) \text { has only one real root } \\ & \Rightarrow \quad f(18)=-233 \\ & \quad f(19)=116 \\ & \Rightarrow \quad f(\alpha)=0 \\ & \Rightarrow \quad \alpha \in(18,19) \end{aligned}$$ For this equation $x \in[16,19)$ $\Rightarrow \quad[x]=16,17,18$ (i) $[x]=16$ $$\begin{aligned} \Rightarrow \quad & 15 x^2+15 x+16=16^3 \\ & (x-16)(x+17)=0 \\ \Rightarrow \quad & x=16 \text { satisfies } \end{aligned}$$ (ii) $[x]=17$ $$\begin{aligned} & \Rightarrow \quad 15 x^2+15 x+16=17^3 \\ & \Rightarrow \quad x=\frac{-15 \pm \sqrt{15^2-4.16 .15}}{30} \\ & \Rightarrow \quad[x]=17 \text { satisfies } \end{aligned}$$ (iii) $[\mathrm{x}]=18$ $\Rightarrow \quad x>19$ and $x < -20$ $\Rightarrow \quad$ No received value of $x$ $\Rightarrow$ Sum of $[x]=16+17=33$.