Question 643589
{{{4x=-5x^2 +2x^3}}}
For any polynomial equation of degree greater than 1, you want one side of the equation to be a zero. So we'll start by subtracting 4x from each side:
{{{ 0=-5x^2 +2x^3-4x}}}<br>
Before we factor, let's put the terms in order from highest exponent to lowest:
{{{ 0= 2x^3-5x^2 -4x}}}
Now we factor. First the greatest common factor , GCF, which is x:
{{{ 0= x(2x^2-5x -4)}}}
This is as far as the expression will factor. From this and the Zero Product property we know that one of these factors must be zero. So
{{{x = 0}}} or {{{2x^2-5x-4 = 0}}}
To find the solutions to the second equation, we will use the Quadratic Formula:
{{{x = (-(-5)+-sqrt((-5)^2-4(2)(-4)))/2(2)}}}
Simplifying...
{{{x = (-(-5)+-sqrt(25-4(2)(-4)))/2(2)}}}
{{{x = (-(-5)+-sqrt(25+32))/2(2)}}}
{{{x = (-(-5)+-sqrt(67))/2(2)}}}
{{{x = (5+-sqrt(67))/4}}}
which is short for:
{{{x = (5+sqrt(67))/4}}} or {{{x = (5-sqrt(67))/4}}}<br>
So the three solutions are:
{{{x = 0}}} or {{{x = (5+sqrt(67))/4}}} or {{{x = (5-sqrt(67))/4}}}