Determine g(x+a)−g(x)
 for the following function.

g(x)=5x^2−3x

We are given the function \( g(x) = 5x^2 - 3x \) and asked to find \( g(x + a) - g(x) \).

### Step 1: Calculate \( g(x + a) \)
We need to substitute \( x + a \) into the function \( g(x) \):
\[
g(x + a) = 5(x + a)^2 - 3(x + a)
\]
Now expand the terms:

\[
g(x + a) = 5(x^2 + 2ax + a^2) - 3(x + a)
\]
Simplifying:
\[
g(x + a) = 5x^2 + 10ax + 5a^2 - 3x - 3a
\]

### Step 2: Find \( g(x + a) - g(x) \)

Now subtract \( g(x) \) from \( g(x + a) \):

\[
g(x + a) - g(x) = \left(5x^2 + 10ax + 5a^2 - 3x - 3a\right) - \left(5x^2 - 3x\right)
\]
Simplify the expression by canceling out the common terms \( 5x^2 \) and \( -3x \):

\[
g(x + a) - g(x) = 10ax + 5a^2 - 3a
\]

### Final Answer:
\[
g(x + a) - g(x) = 10ax + 5a^2 - 3a
\]

This is the result for \( g(x + a) - g(x) \).

Would you like further clarification or more details on any of the steps?

### Related Questions:
1. How would the result change if the function was \( g(x) = 5x^2 + 3x \) instead?
2. What is the significance of \( g(x + a) - g(x) \) in terms of function behavior?
3. How can you apply the same process to a cubic function, such as \( g(x) = x^3 - 2x \)?
4. How would the result differ if you changed the coefficient of \( x^2 \) in the original function?
5. What happens to the expression \( g(x + a) - g(x) \) as \( a \) approaches 0?

### Tip:
Expanding binomials correctly is crucial when dealing with function transformations like \( g(x + a) - g(x) \).

Determine g(x+a)−g(x) for the following function. g(x)=5x^2−3x

Learn how to calculate the difference between g(x + a) and g(x) for the quadratic function g(x) = 5x^2 - 3x. This problem demonstrates key concepts like function transformation and binomial expansion. Suitable for students in Grades 10-12.

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