The difference quotient of a function $f(x)$ is given by the formula:

$\frac{f(x + h) - f(x)}{h}, \quad h \neq 0$

For the function $f(x) = x^2 - 5x + 7$, follow these steps to find the difference quotient:

Step 1: Compute $f(x + h)$

Substitute $x + h$ into $f(x) = x^2 - 5x + 7$:

$f(x + h) = (x + h)^2 - 5(x + h) + 7$

Expand each term:

$(x + h)^2 = x^2 + 2xh + h^2$

$-5(x + h) = -5x - 5h$

So,

$f(x + h) = x^2 + 2xh + h^2 - 5x - 5h + 7$

Step 2: Compute $f(x + h) - f(x)$

Now subtract $f(x) = x^2 - 5x + 7$ from $f(x + h)$:

$f(x + h) - f(x) = \left(x^2 + 2xh + h^2 - 5x - 5h + 7\right) - \left(x^2 - 5x + 7\right)$

Simplify by canceling out the common terms:

$f(x + h) - f(x) = 2xh + h^2 - 5h$

Step 3: Divide by $h$

Now, divide the expression by $h$:

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

Simplify the expression by factoring $h$ out of the numerator:

$\frac{f(x + h) - f(x)}{h} = \frac{h(2x + h - 5)}{h}$

Since $h \neq 0$, we can cancel out $h$:

$\frac{f(x + h) - f(x)}{h} = 2x + h - 5$

Final Answer:

The simplified difference quotient is:

$\boxed{2x + h - 5}$

Do you want more details or have any questions?

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Here are 5 related questions to expand on this topic:

1. How do you find the difference quotient for a cubic function?
2. What happens to the difference quotient as $h \to 0$?
3. How is the difference quotient related to the derivative?
4. How do you compute the difference quotient for a function with a radical expression?
5. Can you compute the difference quotient for $f(x) = 3x^2 - 4x + 1$?

Tip: The difference quotient is a fundamental concept for understanding the derivative in calculus.