Find the Difference Quotient f(x)=-5x^2-7x+6

$f (x) = - 5 x^{2} - 7 x + 6$

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.

$f (x + h) = - 5 {(x + h)}^{2} - 7 (x + h) + 6$

Simplify the result.

Simplify each term.

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

$f (x + h) = - 5 ((x + h) (x + h)) - 7 (x + h) + 6$

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

Apply the distributive property.

$f (x + h) = - 5 (x (x + h) + h (x + h)) - 7 (x + h) + 6$

Apply the distributive property.

$f (x + h) = - 5 (x \cdot x + x h + h (x + h)) - 7 (x + h) + 6$

Apply the distributive property.

$f (x + h) = - 5 (x \cdot x + x h + h x + h \cdot h) - 7 (x + h) + 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = - 5 (x^{2} + x h + h x + h \cdot h) - 7 (x + h) + 6$

Multiply $h$ by $h$ .

$f (x + h) = - 5 (x^{2} + x h + h x + h^{2}) - 7 (x + h) + 6$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = - 5 (x^{2} + h x + h x + h^{2}) - 7 (x + h) + 6$

Add $h x$ and $h x$ .

$f (x + h) = - 5 (x^{2} + 2 h x + h^{2}) - 7 (x + h) + 6$

Apply the distributive property.

$f (x + h) = - 5 x^{2} - 5 (2 h x) - 5 h^{2} - 7 (x + h) + 6$

Multiply $2$ by $- 5$ .

$f (x + h) = - 5 x^{2} - 10 (h x) - 5 h^{2} - 7 (x + h) + 6$

Apply the distributive property.

$f (x + h) = - 5 x^{2} - 10 h x - 5 h^{2} - 7 x - 7 h + 6$

The final answer is $- 5 x^{2} - 10 h x - 5 h^{2} - 7 x - 7 h + 6$ .

$- 5 x^{2} - 10 h x - 5 h^{2} - 7 x - 7 h + 6$

Reorder.

Move $- 7 x$ .

$- 5 x^{2} - 10 h x - 5 h^{2} - 7 h - 7 x + 6$

Move $- 5 x^{2}$ .

$- 10 h x - 5 h^{2} - 5 x^{2} - 7 h - 7 x + 6$

Reorder $- 10 h x$ and $- 5 h^{2}$ .

$- 5 h^{2} - 10 h x - 5 x^{2} - 7 h - 7 x + 6$

Find the components of the definition.

$f (x + h) = - 5 h^{2} - 10 h x - 5 x^{2} - 7 h - 7 x + 6$

$f (x) = - 5 x^{2} - 7 x + 6$

$f (x + h) = - 5 h^{2} - 10 h x - 5 x^{2} - 7 h - 7 x + 6$

$f (x) = - 5 x^{2} - 7 x + 6$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 5 h^{2} - 10 h x - 5 x^{2} - 7 h - 7 x + 6 - (- 5 x^{2} - 7 x + 6)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{- 5 h^{2} - 10 h x - 5 x^{2} - 7 h - 7 x + 6 - (- 5 x^{2}) - (- 7 x) - 1 \cdot 6}{h}$

Simplify.

Multiply $- 5$ by $- 1$ .

$\frac{- 5 h^{2} - 10 h x - 5 x^{2} - 7 h - 7 x + 6 + 5 x^{2} - (- 7 x) - 1 \cdot 6}{h}$

Multiply $- 7$ by $- 1$ .

$\frac{- 5 h^{2} - 10 h x - 5 x^{2} - 7 h - 7 x + 6 + 5 x^{2} + 7 x - 1 \cdot 6}{h}$

Multiply $- 1$ by $6$ .

$\frac{- 5 h^{2} - 10 h x - 5 x^{2} - 7 h - 7 x + 6 + 5 x^{2} + 7 x - 6}{h}$

Add $- 5 x^{2}$ and $5 x^{2}$ .

$\frac{- 5 h^{2} - 10 h x - 7 h - 7 x + 6 + 0 + 7 x - 6}{h}$

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

$\frac{- 5 h^{2} - 10 h x - 7 h - 7 x + 6 + 7 x - 6}{h}$

Add $- 7 x$ and $7 x$ .

$\frac{- 5 h^{2} - 10 h x - 7 h + 0 + 6 - 6}{h}$

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

$\frac{- 5 h^{2} - 10 h x - 7 h + 6 - 6}{h}$

Subtract $6$ from $6$ .

$\frac{- 5 h^{2} - 10 h x - 7 h + 0}{h}$

Add $- 5 h^{2} - 10 h x - 7 h$ and $0$ .

$\frac{- 5 h^{2} - 10 h x - 7 h}{h}$

Factor $h$ out of $- 5 h^{2} - 10 h x - 7 h$ .

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

$\frac{h (- 5 h) - 10 h x - 7 h}{h}$

Factor $h$ out of $- 10 h x$ .

$\frac{h (- 5 h) + h (- 10 x) - 7 h}{h}$

Factor $h$ out of $- 7 h$ .

$\frac{h (- 5 h) + h (- 10 x) + h \cdot - 7}{h}$

Factor $h$ out of $h (- 5 h) + h (- 10 x)$ .

$\frac{h (- 5 h - 10 x) + h \cdot - 7}{h}$

Factor $h$ out of $h (- 5 h - 10 x) + h \cdot - 7$ .

$\frac{h (- 5 h - 10 x - 7)}{h}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{h (- 5 h - 10 x - 7)}{h}$

Divide $- 5 h - 10 x - 7$ by $1$ .

$- 5 h - 10 x - 7$

Reorder $- 5 h$ and $- 10 x$ .

$- 10 x - 5 h - 7$

Step 5

Find the Difference Quotient f(x)=-5x^2-7x+6

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