Find the Difference Quotient f(x)=-7x^2-5x-8

$f (x) = - 7 x^{2} - 5 x - 8$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

Evaluate the function at $x = x + h$ .

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = - 7 {(x + h)}^{2} - 5 (x + h) - 8$

Simplify the result.

Simplify each term.

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$f (x + h) = - 7 ((x + h) (x + h)) - 5 (x + h) - 8$

Expand $(x + h) (x + h)$ using the FOIL Method.

Apply the distributive property.

$f (x + h) = - 7 (x (x + h) + h (x + h)) - 5 (x + h) - 8$

Apply the distributive property.

$f (x + h) = - 7 (x \cdot x + x h + h (x + h)) - 5 (x + h) - 8$

Apply the distributive property.

$f (x + h) = - 7 (x \cdot x + x h + h x + h \cdot h) - 5 (x + h) - 8$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = - 7 (x^{2} + x h + h x + h \cdot h) - 5 (x + h) - 8$

Multiply $h$ by $h$ .

$f (x + h) = - 7 (x^{2} + x h + h x + h^{2}) - 5 (x + h) - 8$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = - 7 (x^{2} + h x + h x + h^{2}) - 5 (x + h) - 8$

Add $h x$ and $h x$ .

$f (x + h) = - 7 (x^{2} + 2 h x + h^{2}) - 5 (x + h) - 8$

Apply the distributive property.

$f (x + h) = - 7 x^{2} - 7 (2 h x) - 7 h^{2} - 5 (x + h) - 8$

Multiply $2$ by $- 7$ .

$f (x + h) = - 7 x^{2} - 14 (h x) - 7 h^{2} - 5 (x + h) - 8$

Apply the distributive property.

$f (x + h) = - 7 x^{2} - 14 h x - 7 h^{2} - 5 x - 5 h - 8$

The final answer is $- 7 x^{2} - 14 h x - 7 h^{2} - 5 x - 5 h - 8$ .

$- 7 x^{2} - 14 h x - 7 h^{2} - 5 x - 5 h - 8$

Reorder.

Move $- 5 x$ .

$- 7 x^{2} - 14 h x - 7 h^{2} - 5 h - 5 x - 8$

Move $- 7 x^{2}$ .

$- 14 h x - 7 h^{2} - 7 x^{2} - 5 h - 5 x - 8$

Reorder $- 14 h x$ and $- 7 h^{2}$ .

$- 7 h^{2} - 14 h x - 7 x^{2} - 5 h - 5 x - 8$

Find the components of the definition.

$f (x + h) = - 7 h^{2} - 14 h x - 7 x^{2} - 5 h - 5 x - 8$

$f (x) = - 7 x^{2} - 5 x - 8$

$f (x + h) = - 7 h^{2} - 14 h x - 7 x^{2} - 5 h - 5 x - 8$

$f (x) = - 7 x^{2} - 5 x - 8$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 7 h^{2} - 14 h x - 7 x^{2} - 5 h - 5 x - 8 - (- 7 x^{2} - 5 x - 8)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{- 7 h^{2} - 14 h x - 7 x^{2} - 5 h - 5 x - 8 - (- 7 x^{2}) - (- 5 x) - - 8}{h}$

Simplify.

Multiply $- 7$ by $- 1$ .

$\frac{- 7 h^{2} - 14 h x - 7 x^{2} - 5 h - 5 x - 8 + 7 x^{2} - (- 5 x) - - 8}{h}$

Multiply $- 5$ by $- 1$ .

$\frac{- 7 h^{2} - 14 h x - 7 x^{2} - 5 h - 5 x - 8 + 7 x^{2} + 5 x - - 8}{h}$

Multiply $- 1$ by $- 8$ .

$\frac{- 7 h^{2} - 14 h x - 7 x^{2} - 5 h - 5 x - 8 + 7 x^{2} + 5 x + 8}{h}$

Add $- 7 x^{2}$ and $7 x^{2}$ .

$\frac{- 7 h^{2} - 14 h x - 5 h - 5 x - 8 + 0 + 5 x + 8}{h}$

Add $- 7 h^{2}$ and $0$ .

$\frac{- 7 h^{2} - 14 h x - 5 h - 5 x - 8 + 5 x + 8}{h}$

Add $- 5 x$ and $5 x$ .

$\frac{- 7 h^{2} - 14 h x - 5 h + 0 - 8 + 8}{h}$

Add $- 7 h^{2}$ and $0$ .

$\frac{- 7 h^{2} - 14 h x - 5 h - 8 + 8}{h}$

Add $- 8$ and $8$ .

$\frac{- 7 h^{2} - 14 h x - 5 h + 0}{h}$

Add $- 7 h^{2} - 14 h x - 5 h$ and $0$ .

$\frac{- 7 h^{2} - 14 h x - 5 h}{h}$

Factor $h$ out of $- 7 h^{2} - 14 h x - 5 h$ .

Factor $h$ out of $- 7 h^{2}$ .

$\frac{h (- 7 h) - 14 h x - 5 h}{h}$

Factor $h$ out of $- 14 h x$ .

$\frac{h (- 7 h) + h (- 14 x) - 5 h}{h}$

Factor $h$ out of $- 5 h$ .

$\frac{h (- 7 h) + h (- 14 x) + h \cdot - 5}{h}$

Factor $h$ out of $h (- 7 h) + h (- 14 x)$ .

$\frac{h (- 7 h - 14 x) + h \cdot - 5}{h}$

Factor $h$ out of $h (- 7 h - 14 x) + h \cdot - 5$ .

$\frac{h (- 7 h - 14 x - 5)}{h}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{h (- 7 h - 14 x - 5)}{h}$

Divide $- 7 h - 14 x - 5$ by $1$ .

$- 7 h - 14 x - 5$

Reorder $- 7 h$ and $- 14 x$ .

$- 14 x - 7 h - 5$

Step 5

Find the Difference Quotient f(x)=-7x^2-5x-8

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