Question

Find all values of the parameter for which each of the roots of the equation x^3 - 3x - a = 0 not less than (a/10) + (6/5). In response, write down the sum of the integer values of the parameter a such that their modulus is less than 61.

Answer 1

Solution-1:

1) We should simply the given equation:

$x(x^2-3)=a$

in this case, we assume that:

$x_1=a\;$

$x^2-3=1$ from this we obtain the remaining roots of the equation:

$x_2=2\;;\;x_3=-2$

Let`s back to the question:

I)

$\frac{a}{10}+\frac{6}{5}\geq-2$

$a\geq-32$

II)

$a+|-2|+2\lt61$

$a\lt57$

Answer:$a\;\epsilon [-32,57)$

Solution-2:

1) We should simply the given equation:

$x(x^2-3)=a$

in this case, we assume that:

$x_1=1\;$

$x^2-3=a$ from this we obtain the remaining roots of the equation:

$x_2=\sqrt{\smash[b]{a+3}}\;;\;x_3=-\sqrt{\smash[b]{a+3}}$

Let`s back to the question:

I)

$\frac{a}{10}+\frac{6}{5}\geq-\sqrt{\smash[b]{a+3}}$

$a\;\epsilon[-\infty,\infty]$ but the domain of $x_2\;,x_3$ should be $a\geq\;-3$

II)

$1+|-\sqrt{\smash[b]{a+3}}|+|\sqrt{\smash[b]{a+3}}|\lt61$

$a\lt897$

Answer:

$a\;\epsilon [-3,897)$