To solve the equation [tex]\(-5x(7x - 4)(3x + 1)^2 = 0\)[/tex], we need to factorize and solve for [tex]\(x\)[/tex]. Let's break it down step-by-step.
### Step 1: Understand the Equation
The equation is [tex]\(-5x(7x - 4)(3x + 1)^2 = 0\)[/tex]. This equation is a product of three factors: [tex]\(-5x\)[/tex], [tex]\((7x - 4)\)[/tex], and [tex]\((3x + 1)^2\)[/tex].
### Step 2: Apply the Zero Product Property
According to the Zero Product Property, if a product of several factors is zero, then at least one of the factors must be zero. Thus, we can set each factor to zero and solve for [tex]\(x\)[/tex].
1. [tex]\(-5x = 0\)[/tex]
2. [tex]\(7x - 4 = 0\)[/tex]
3. [tex]\((3x + 1)^2 = 0\)[/tex]
### Step 3: Solve Each Factor Individually
#### Factor 1: [tex]\(-5x = 0\)[/tex]
To solve for [tex]\(x\)[/tex], we divide both sides by [tex]\(-5\)[/tex]:
[tex]\[
x = 0
\][/tex]
#### Factor 2: [tex]\(7x - 4 = 0\)[/tex]
To solve for [tex]\(x\)[/tex], we isolate [tex]\(x\)[/tex]:
[tex]\[
7x - 4 = 0 \\
7x = 4 \\
x = \frac{4}{7}
\][/tex]
#### Factor 3: [tex]\((3x + 1)^2 = 0\)[/tex]
Since [tex]\((3x + 1)^2\)[/tex] is a square term, it is zero when the term inside the square is zero:
[tex]\[
3x + 1 = 0 \\
3x = -1 \\
x = -\frac{1}{3}
\][/tex]
### Step 4: Compile the Solutions
The solutions to the equation [tex]\(-5x(7x - 4)(3x + 1)^2 = 0\)[/tex] are:
[tex]\[
x = 0, \quad x = \frac{4}{7}, \quad x = -\frac{1}{3}
\][/tex]
### Conclusion
The values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(-5x(7x - 4)(3x + 1)^2 = 0\)[/tex] are:
[tex]\[
x = 0, \quad x = \frac{4}{7}, \quad x = -\frac{1}{3}
\][/tex]
These are the points where the given equation equals zero.
