Questions: Solve the equation (2 k=-9 k^3-9 k^2) Answer: (k=) Give your answers as integers or reduced fractions, separated by commas.

$Solve the equation (2 k=-9 k^3-9 k^2) Answer: (k=) Give your answers as integers or reduced fractions, separated by commas.$

Transcript text: Solve the equation $2 k=-9 k^{3}-9 k^{2}$ Answer: $k=$ $\square$ Give your answers as integers or reduced fractions, separated by commas.

Solution

Solution Steps

To solve the equation $2k = -9k^3 - 9k^2$, we need to rearrange it into a standard polynomial form and then find the roots of the polynomial. This involves moving all terms to one side of the equation and then using a root-finding method to solve for $k$.

Step 1: Rearrange the Equation

First, we start with the given equation: \[ 2k = -9k^3 - 9k^2 \]

We rearrange all terms to one side to form a standard polynomial equation: \[ 9k^3 + 9k^2 + 2k = 0 \]

Step 2: Factor the Polynomial

Next, we factor out the common term $ k $: \[ k(9k^2 + 9k + 2) = 0 \]

Step 3: Solve for $ k $

We solve for $ k $ by setting each factor to zero:

$ k = 0 $
Solve the quadratic equation $ 9k^2 + 9k + 2 = 0 $

Using the quadratic formula $ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $, where $ a = 9 $, $ b = 9 $, and $ c = 2 $: \[ k = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 9 \cdot 2}}{2 \cdot 9} \] \[ k = \frac{-9 \pm \sqrt{81 - 72}}{18} \] \[ k = \frac{-9 \pm \sqrt{9}}{18} \] \[ k = \frac{-9 \pm 3}{18} \]

This gives us two solutions: \[ k = \frac{-9 + 3}{18} = \frac{-6}{18} = -\frac{1}{3} \] \[ k = \frac{-9 - 3}{18} = \frac{-12}{18} = -\frac{2}{3} \]