Question

5x^3=6x^2-45x-54

Answer 1

Set the polynomial equal to zero; commute the squared and first degree terms\[5x^3+45x-6x^2-54=0\]Factor GCF from terms 1 & 2 and 3 & 4\[5x(x^2+9)-6(x^2+9)=0\]Factor out binomial as GCF\[(x^2+9)(5x-6)=0\]The first factor produces two imaginary roots\[x^2+9=0\]\[x^2=-9\]\[x=\pm3i\]The second factor yields one real root\[5x-6=0\]\[x=\frac{6}{5}\]If your directions are "find all the roots" the solution set is\[\left\{ -3i, 3i, \frac{5}{6} \right\}\]If your directions are to "find all the real roots" the solution set is\[\left\{ \frac{5}{6} \right\}\]