Question

2 g(x)=3x² - 3x

Explanation:

Since the problem statement for the function \( g(x) = 3x^2 - 3x \) is incomplete (e.g., we don't know what is being asked about this function, like finding the vertex, roots, derivative, integral, or something else), we can't provide a solution yet. If you want to find the roots (x-intercepts) of the function, for example, here's how we could do it:

Step 1: Set \( g(x) = 0 \)

We start by setting the function equal to zero because the roots of a function are the values of \( x \) for which \( g(x) = 0 \). So we have the equation:
\( 3x^2 - 3x = 0 \)

Step 2: Factor out the common term

We can factor out a \( 3x \) from both terms on the left - hand side of the equation. Using the distributive property in reverse (\( ab+ac = a(b + c) \), here \( a = 3x \), \( b=x \), \( c=- 1 \)), we get:
\( 3x(x - 1)=0 \)

Step 3: Use the zero - product property

The zero - product property states that if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \) (or both). So from \( 3x(x - 1)=0 \), we have two cases:
Case 1: \( 3x=0 \)
Dividing both sides by 3, we get \( x = 0 \).
Case 2: \( x - 1=0 \)
Adding 1 to both sides, we get \( x=1 \).

If this is not the operation you wanted to perform on the function \( g(x)=3x^2 - 3x \) (such as finding the vertex, derivative, integral, etc.), please provide more details about what you need to do with the function.

Answer:

Step 1: Set \( g(x) = 0 \)

We start by setting the function equal to zero because the roots of a function are the values of \( x \) for which \( g(x) = 0 \). So we have the equation:
\( 3x^2 - 3x = 0 \)

Step 2: Factor out the common term

We can factor out a \( 3x \) from both terms on the left - hand side of the equation. Using the distributive property in reverse (\( ab+ac = a(b + c) \), here \( a = 3x \), \( b=x \), \( c=- 1 \)), we get:
\( 3x(x - 1)=0 \)

Step 3: Use the zero - product property

The zero - product property states that if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \) (or both). So from \( 3x(x - 1)=0 \), we have two cases:
Case 1: \( 3x=0 \)
Dividing both sides by 3, we get \( x = 0 \).
Case 2: \( x - 1=0 \)
Adding 1 to both sides, we get \( x=1 \).

If this is not the operation you wanted to perform on the function \( g(x)=3x^2 - 3x \) (such as finding the vertex, derivative, integral, etc.), please provide more details about what you need to do with the function.