given the function f(x)=2x²+5x-3 find f(x+h)-f(x)/h

To solve \( \frac{f(x+h) - f(x)}{h} \) for the function \( f(x) = 2x^2 + 5x - 3 \), let's go step by step.

### Step 1: Find \( f(x+h) \)
We need to substitute \( x + h \) into the function \( f(x) \).

\[
f(x+h) = 2(x+h)^2 + 5(x+h) - 3
\]

Now expand the terms:
\[
= 2(x^2 + 2xh + h^2) + 5(x + h) - 3
\]
\[
= 2x^2 + 4xh + 2h^2 + 5x + 5h - 3
\]

### Step 2: Subtract \( f(x) \)
Now subtract \( f(x) = 2x^2 + 5x - 3 \) from \( f(x+h) \):

\[
f(x+h) - f(x) = \left( 2x^2 + 4xh + 2h^2 + 5x + 5h - 3 \right) - \left( 2x^2 + 5x - 3 \right)
\]
Simplify by canceling out the common terms:
\[
f(x+h) - f(x) = 4xh + 2h^2 + 5h
\]

### Step 3: Divide by \( h \)
Now, divide the result by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{4xh + 2h^2 + 5h}{h}
\]

Factor out \( h \) in the numerator:
\[
= \frac{h(4x + 2h + 5)}{h}
\]

Since \( h \neq 0 \), we can cancel out \( h \):

\[
= 4x + 2h + 5
\]

### Final Answer:
\[
\frac{f(x+h) - f(x)}{h} = 4x + 2h + 5
\]

Would you like a more detailed explanation or have any questions?

Here are some related questions to practice:
1. What is the limit of \( \frac{f(x+h) - f(x)}{h} \) as \( h \to 0 \)?
2. Can you find the derivative of \( f(x) \)?
3. What is the equation of the tangent line to the graph of \( f(x) \) at \( x = 2 \)?
4. How would you compute \( \frac{f(x+2) - f(x)}{2} \)?
5. What does \( f'(x) \) tell you about the graph of \( f(x) \)?

**Tip:** When simplifying, always look for common terms to cancel out—it makes calculations much easier!

Learn how to find the difference quotient (f(x+h) - f(x))/h for the function f(x) = 2x² + 5x - 3 with a step-by-step solution. Understand key algebraic concepts, including polynomial expansion and simplification. Suitable for students in Grades 9-12.

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