solve the equation y^3+2y^2=9y+18

Question

solve the equation y^3+2y^2=9y+18

Answer 1

     y³ + 2y² = 9y + 18

First, set the equation equal to 0 by subtracting the right-hand side from both sides of the equation:

     y³ + 2y² - 9y - 18 = 0

We solve this polynomial equation by factoring. To do so, consider the first two terms and last two terms separately and look to see if they contain a greatest common factor that you can factor out of each set separately:

     y³ + 2y²   ==>   the gcf here is 'y²'   ==>   y²(y + 2)

    -9y - 18   ==>   the gcf here is '-9'   ==>   -9(y + 2)

Thus,

     y³ + 2y² - 9y - 18 = 0

     [y²(y + 2)] + [-9(y + 2)] = 0

Notice that there is now a greatest common factor among these two terms, that being 'y + 2'. So after factoring out this gcf, we arrive at the following:

     (y + 2)(y² - 9) = 0

By the zero product property, we can now set each binomial equal to 0 and solve for y:

     y + 2 = 0    ==>    subtract 2 from both sides of the equation

          y = -2

     y² - 9 = 0    ==>    add 9 to both sides of the equation

         y² = 9    ==>    take the square root of both sides of the equation

      √y² = ±√9

         y = ±3 

Thus, there are 3 solutions for this equation. Those being,  y = -2 ,  y = -3 ,  and  y = 3.