find the real or imaginary solutions of each equation by factoring.
3x^3+3x^2=27x
*please show work so i can learn how to do this.
first, put into descending powers:
3x^3+3x^2-27x = 0
Now factor out 3x, leaving
3x(x^2+x-9) = 0
so, clearly, x=0 is one solution
use the quadratic formula to find the others. You know they are real, since the discriminant is positive.
To find the real or imaginary solutions of the equation, we will start by factoring.
Step 1: Set the equation equal to zero by subtracting 27x from both sides:
3x^3 + 3x^2 - 27x = 0
Step 2: Factor out the greatest common factor, which in this case is 3x:
3x(x^2 + x - 9) = 0
Now we have factored out the greatest common factor. We can see that the equation can be solved by setting each factor equal to zero.
Step 3: Set the first factor, 3x, equal to zero and solve for x:
3x = 0
x = 0
Step 4: Set the second factor, x^2 + x - 9, equal to zero and solve for x. We can use factoring or the quadratic formula in this step.
Factoring method:
The quadratic expression x^2 + x - 9 can be factored as follows:
(x + 3)(x - 3) = 0
Setting each factor equal to zero:
x + 3 = 0
x = -3
x - 3 = 0
x = 3
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 1, and c = -9. Substituting these values into the formula:
x = (-1 ± √(1^2 - 4(1)(-9))) / (2(1))
x = (-1 ± √(1 + 36)) / 2
x = (-1 ± √37) / 2
So the two solutions using the quadratic formula are x = (-1 + √37) / 2 and x = (-1 - √37) / 2.
Therefore, the real or imaginary solutions of the equation 3x^3 + 3x^2 = 27x, after factoring, are:
x = 0, -3, and the two solutions using the quadratic formula: x = (-1 + √37) / 2 and x = (-1 - √37) / 2.