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:

Correct answer

\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