Find the Difference Quotient f(x)=10x^2+5x-12

$f (x) = 10 x^{2} + 5 x - 12$

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.

$f (x + h) = 10 {(x + h)}^{2} + 5 (x + h) - 12$

Simplify the result.

Simplify each term.

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

$f (x + h) = 10 ((x + h) (x + h)) + 5 (x + h) - 12$

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

Apply the distributive property.

$f (x + h) = 10 (x (x + h) + h (x + h)) + 5 (x + h) - 12$

Apply the distributive property.

$f (x + h) = 10 (x \cdot x + x h + h (x + h)) + 5 (x + h) - 12$

Apply the distributive property.

$f (x + h) = 10 (x \cdot x + x h + h x + h \cdot h) + 5 (x + h) - 12$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x + h) = 10 (x^{2} + x h + h x + h \cdot h) + 5 (x + h) - 12$

Multiply $h$ by $h$ .

$f (x + h) = 10 (x^{2} + x h + h x + h^{2}) + 5 (x + h) - 12$

Add $x h$ and $h x$ .

Reorder $x$ and $h$ .

$f (x + h) = 10 (x^{2} + h x + h x + h^{2}) + 5 (x + h) - 12$

Add $h x$ and $h x$ .

$f (x + h) = 10 (x^{2} + 2 h x + h^{2}) + 5 (x + h) - 12$

Apply the distributive property.

$f (x + h) = 10 x^{2} + 10 (2 h x) + 10 h^{2} + 5 (x + h) - 12$

Multiply $2$ by $10$ .

$f (x + h) = 10 x^{2} + 20 (h x) + 10 h^{2} + 5 (x + h) - 12$

Apply the distributive property.

$f (x + h) = 10 x^{2} + 20 h x + 10 h^{2} + 5 x + 5 h - 12$

The final answer is $10 x^{2} + 20 h x + 10 h^{2} + 5 x + 5 h - 12$ .

$10 x^{2} + 20 h x + 10 h^{2} + 5 x + 5 h - 12$

Reorder.

Move $5 x$ .

$10 x^{2} + 20 h x + 10 h^{2} + 5 h + 5 x - 12$

Move $10 x^{2}$ .

$20 h x + 10 h^{2} + 10 x^{2} + 5 h + 5 x - 12$

Reorder $20 h x$ and $10 h^{2}$ .

$10 h^{2} + 20 h x + 10 x^{2} + 5 h + 5 x - 12$

Find the components of the definition.

$f (x + h) = 10 h^{2} + 20 h x + 10 x^{2} + 5 h + 5 x - 12$

$f (x) = 10 x^{2} + 5 x - 12$

$f (x + h) = 10 h^{2} + 20 h x + 10 x^{2} + 5 h + 5 x - 12$

$f (x) = 10 x^{2} + 5 x - 12$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{10 h^{2} + 20 h x + 10 x^{2} + 5 h + 5 x - 12 - (10 x^{2} + 5 x - 12)}{h}$

Step 4

Simplify.

Simplify the numerator.

Apply the distributive property.

$\frac{10 h^{2} + 20 h x + 10 x^{2} + 5 h + 5 x - 12 - (10 x^{2}) - (5 x) - - 12}{h}$

Simplify.

Multiply $10$ by $- 1$ .

$\frac{10 h^{2} + 20 h x + 10 x^{2} + 5 h + 5 x - 12 - 10 x^{2} - (5 x) - - 12}{h}$

Multiply $5$ by $- 1$ .

$\frac{10 h^{2} + 20 h x + 10 x^{2} + 5 h + 5 x - 12 - 10 x^{2} - 5 x - - 12}{h}$

Multiply $- 1$ by $- 12$ .

$\frac{10 h^{2} + 20 h x + 10 x^{2} + 5 h + 5 x - 12 - 10 x^{2} - 5 x + 12}{h}$

Subtract $10 x^{2}$ from $10 x^{2}$ .

$\frac{10 h^{2} + 20 h x + 5 h + 5 x - 12 + 0 - 5 x + 12}{h}$

Add $10 h^{2}$ and $0$ .

$\frac{10 h^{2} + 20 h x + 5 h + 5 x - 12 - 5 x + 12}{h}$

Subtract $5 x$ from $5 x$ .

$\frac{10 h^{2} + 20 h x + 5 h + 0 - 12 + 12}{h}$

Add $10 h^{2}$ and $0$ .

$\frac{10 h^{2} + 20 h x + 5 h - 12 + 12}{h}$

Add $- 12$ and $12$ .

$\frac{10 h^{2} + 20 h x + 5 h + 0}{h}$

Add $10 h^{2} + 20 h x + 5 h$ and $0$ .

$\frac{10 h^{2} + 20 h x + 5 h}{h}$

Factor $5 h$ out of $10 h^{2} + 20 h x + 5 h$ .

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

$\frac{5 h (2 h) + 20 h x + 5 h}{h}$

Factor $5 h$ out of $20 h x$ .

$\frac{5 h (2 h) + 5 h (4 x) + 5 h}{h}$

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

$\frac{5 h (2 h) + 5 h (4 x) + 5 h \cdot 1}{h}$

Factor $5 h$ out of $5 h (2 h) + 5 h (4 x)$ .

$\frac{5 h (2 h + 4 x) + 5 h \cdot 1}{h}$

Factor $5 h$ out of $5 h (2 h + 4 x) + 5 h \cdot 1$ .

$\frac{5 h (2 h + 4 x + 1)}{h}$

Simplify terms.

Cancel the common factor of $h$ .

Cancel the common factor.

$\frac{5 h (2 h + 4 x + 1)}{h}$

Divide $5 (2 h + 4 x + 1)$ by $1$ .

$5 (2 h + 4 x + 1)$

Apply the distributive property.

$5 (2 h) + 5 (4 x) + 5 \cdot 1$

Simplify.

Multiply $2$ by $5$ .

$10 h + 5 (4 x) + 5 \cdot 1$

Multiply $4$ by $5$ .

$10 h + 20 x + 5 \cdot 1$

Multiply $5$ by $1$ .

$10 h + 20 x + 5$

Reorder $10 h$ and $20 x$ .

$20 x + 10 h + 5$

Step 5

Find the Difference Quotient f(x)=10x^2+5x-12

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