Solve the equation.

44ccubed3plus+2424csquared2negative 108−108cequals=0

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Part 1

The solution set is 

StartSet nothing EndSetenter your response here.

The equation you provided seems to describe a cubic equation involving the variable \( c \). Based on the text, I will interpret the equation as follows:

\[
44c^3 + 24c^2 - 108c = 0
\]

This is a cubic equation that can be solved by factoring. Let's solve it step-by-step:

### Step 1: Factor out the greatest common factor (GCF)
The greatest common factor of the terms is \( 4c \). So, we factor out \( 4c \):

\[
4c(11c^2 + 6c - 27) = 0
\]

### Step 2: Solve the first factor
The first factor is \( 4c = 0 \), which gives:

\[
c = 0
\]

### Step 3: Solve the quadratic equation
Now, we solve the quadratic equation \( 11c^2 + 6c - 27 = 0 \). We can apply the quadratic formula:

\[
c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the quadratic \( 11c^2 + 6c - 27 = 0 \), the coefficients are:
- \( a = 11 \)
- \( b = 6 \)
- \( c = -27 \)

Substituting these values into the quadratic formula:

\[
c = \frac{-6 \pm \sqrt{6^2 - 4(11)(-27)}}{2(11)}
\]
\[
c = \frac{-6 \pm \sqrt{36 + 1188}}{22}
\]
\[
c = \frac{-6 \pm \sqrt{1224}}{22}
\]
\[
c = \frac{-6 \pm 34.98}{22}
\]

### Step 4: Find the roots
Now, we find the two possible solutions:

\[
c_1 = \frac{-6 + 34.98}{22} \approx \frac{28.98}{22} \approx 1.32
\]
\[
c_2 = \frac{-6 - 34.98}{22} \approx \frac{-40.98}{22} \approx -1.86
\]

### Step 5: Combine the solutions
The solutions to the equation are:

\[
c = 0, \, c \approx 1.32, \, c \approx -1.86
\]

Thus, the solution set is approximately:

\[
\{ 0, 1.32, -1.86 \}
\]

Would you like further details or clarifications? Here are 5 related questions to explore further:

1. How is the quadratic formula derived?
2. How do we determine the greatest common factor (GCF) when factoring equations?
3. Can cubic equations always be factored into simpler parts?
4. What are the possible types of roots in a cubic equation?
5. How do we check if our solutions are correct?

**Tip:** Always check your solutions by plugging them back into the original equation to ensure they satisfy it.

Solve the equation: 44c^3 + 24c^2 - 108c = 0

This problem involves solving the cubic equation 44c^3 + 24c^2 - 108c = 0 using factoring and the quadratic formula. Detailed steps include factoring out the GCF and applying the quadratic formula to find the roots. Suitable for students in Grades 10-12.

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