Question

I'm studying algebra and I'm stuck on a problem that involves factoring. Can you find a solution to the equation 6x^3 + 2x^2 − 15x − 5 = 0 using the method of factoring by grouping?

Answer 1

To solve the equation \( 6x^3 + 2x^2 - 15x - 5 = 0 \) using the method of factoring by grouping, we first need to group the terms in a way that allows us to factor them effectively.

We can start by rearranging the expression, but let's first split it into two groups:

\[
(6x^3 + 2x^2) + (-15x - 5)
\]

Now we can factor out the greatest common factor (GCF) from each group:

1. From the first group \( 6x^3 + 2x^2 \), the GCF is \( 2x^2 \):
\[
2x^2(3x + 1)
\]

2. From the second group \( -15x - 5 \), the GCF is \( -5 \):
\[
-5(3x + 1)
\]

Now we rewrite the expression using these factored forms:

\[
2x^2(3x + 1) - 5(3x + 1)
\]

Notice that both groups now contain the common factor \( (3x + 1) \). We can factor \( (3x + 1) \) out:

\[
(3x + 1)(2x^2 - 5) = 0
\]

Now, we can set each factor equal to zero:

1. For \( 3x + 1 = 0 \):
\[
3x = -1 \quad \Rightarrow \quad x = -\frac{1}{3}
\]

2. For \( 2x^2 - 5 = 0 \):
\[
2x^2 = 5 \quad \Rightarrow \quad x^2 = \frac{5}{2} \quad \Rightarrow \quad x = \pm\sqrt{\frac{5}{2}} = \pm\frac{\sqrt{10}}{2}
\]

Thus, the solutions for the equation \( 6x^3 + 2x^2 - 15x - 5 = 0 \) are:

\[
x = -\frac{1}{3}, \quad x = \frac{\sqrt{10}}{2}, \quad x = -\frac{\sqrt{10}}{2}
\]