2022-01-23

$3{x}^{3}=24$

fionaluvsyou0x

Expert

Step 1
May be you know this already but its

oferenteoo

Expert

Step 1
Hints
$3{x}^{3}=24⇔{x}^{3}-8=0$
$\frac{{x}^{3}-8}{x-2}={x}^{2}+2x+4$
Added: The second equation means that
${x}^{3}-8=\left(x-2\right)\left({x}^{2}+2x+4\right)$
Thus we have
${x}^{3}-8=0⇔\left(x-2\right)\left({x}^{3}+2x+4\right)=0$
whose solutions are the values of x that make the factor $x-2$ or the factor ${x}^{2}+2x+4$ equal to 0. The equation $x-2=0$ has the solution $x=2$. The solutions of the equation ${x}^{2}+2x+4=0$ may be found by the quadratic formula or as follows:
${x}^{2}+2x+4=0⇔{\left(x+1\right)}^{2}+3=0⇔{\left(x+1\right)}^{2}=-3$
$⇔x+1=±\sqrt{-3}=±\sqrt{3}i$
$⇔x=-1±\sqrt{3}i$
So
${x}^{3}-8=0⇔x=2$ or $x=-1±\sqrt{3}i$

RizerMix

Expert

factor using polynomial division: $\frac{{x}^{3}-8}{x-2}=\left({x}^{2}+2x+4\right)$ To solve this, we can quickly complete the square as follows: Notice that we are setting ${x}^{2}+2x+4=0$ $\left(x+1{\right)}^{2}+3=0$ $\left(x+1{\right)}^{2}=-3$ $x+1=±i\sqrt{3}$ $x=-1±i\sqrt{3}$ and we have our answer. I find this slightly faster than using the quadratic formula.