Find all real solutions: $36x^3 + 6x^2 = 9x$

Question

I am having trouble finding all the real solutions to this problem. I do not undertand how to solve it, and I have a test on Polynomials tomorrow. The problem is "Find all real solutions using your calculator. Round to the nearest hundredth: $36x^3 + 6x^2 = 9x$"

Answer 1

You can factor the equation as follows. $$36x^{3} + 6x^{2} - 9x = 0$$ $$3x\left(12x^{2} + 2x - 3\right) = 0.$$

From this, you know that one solution is $\boxed{x = 0}.$ We must now find the solutions to $12x^{2} + 2x - 3 = 0.$ As this is not factorable, we can use the quadratic formula to find the others. We do this as follows. $$x = \frac{-2 \pm \sqrt{4 + 144}}{24}$$ $$= \frac{-2 \pm \sqrt{148}}{24}$$ $$= \frac{-2 \pm 2\sqrt{37}}{24}$$ $$= \boxed{-\frac{1}{12} \pm \frac{\sqrt{37}}{12}}.$$

I leave you to find the decimal approximations.

Answer 2

Rewrite as $$36x^3+6x^2-9x=0$$

Note that this is the same as $$ x(36x^2+6x-9)=0 $$

So $0$ is a solution, and also the values for which $$36x^2+6x-9=0$$ are solutions. You can find the latter with the quadratic formula.