Question

1. List all elements in A. A={xϵN:(3x2-x)<55}

Answer 1

Let us list all elements in $A=\{x\in\N:(3x^2-x)<55\}.$

For this let us solve the inequality $3x^2-x<55.$

The discriminant of the equation $3x^2-x-55=0$ is equal to $D=1+3\cdot4\cdot 55=661,$ and hence the equation has the solutions $x_1=\frac{1-\sqrt{661}}{6}\approx -4.11$ and $x_2=\frac{1+\sqrt{661}}{6}\approx 4.45$

It follows that the real solutions of the inequlity $3x^2-x<55$ belong to the set $(\frac{1-\sqrt{661}}{6},\frac{1+\sqrt{661}}{6}).$

This set contains the following natural numbers: $1,2,3,4.$

Consequently, $A=\{1,2,3,4\}.$