Algebra Examples

$f (x) = 2.9 + 5.8 x - 2.4 x^{2}$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

$f (x + h) = 2.9 + 5.8 (x + h) - 2.4 {(x + h)}^{2}$

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Simplify each term.

Tap for more steps...

Step 2.1.2.1.1

Apply the distributive property.

$f (x + h) = 2.9 + 5.8 x + 5.8 h - 2.4 {(x + h)}^{2}$

Step 2.1.2.1.2

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

$f (x + h) = 2.9 + 5.8 x + 5.8 h - 2.4 ((x + h) (x + h))$

Step 2.1.2.1.3

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

Tap for more steps...

Step 2.1.2.1.3.1

Apply the distributive property.

$f (x + h) = 2.9 + 5.8 x + 5.8 h - 2.4 (x (x + h) + h (x + h))$

Step 2.1.2.1.3.2

Apply the distributive property.

$f (x + h) = 2.9 + 5.8 x + 5.8 h - 2.4 (x \cdot x + x h + h (x + h))$

Step 2.1.2.1.3.3

Apply the distributive property.

$f (x + h) = 2.9 + 5.8 x + 5.8 h - 2.4 (x \cdot x + x h + h x + h \cdot h)$

Step 2.1.2.1.4

Simplify and combine like terms.

Tap for more steps...

Step 2.1.2.1.4.1

Simplify each term.

Tap for more steps...

Step 2.1.2.1.4.1.1

Multiply $x$ by $x$ .

$f (x + h) = 2.9 + 5.8 x + 5.8 h - 2.4 (x^{2} + x h + h x + h \cdot h)$

Step 2.1.2.1.4.1.2

Multiply $h$ by $h$ .

$f (x + h) = 2.9 + 5.8 x + 5.8 h - 2.4 (x^{2} + x h + h x + h^{2})$

Step 2.1.2.1.4.2

Add $x h$ and $h x$ .

Tap for more steps...

Step 2.1.2.1.4.2.1

Reorder $x$ and $h$ .

$f (x + h) = 2.9 + 5.8 x + 5.8 h - 2.4 (x^{2} + h x + h x + h^{2})$

Step 2.1.2.1.4.2.2

Add $h x$ and $h x$ .

$f (x + h) = 2.9 + 5.8 x + 5.8 h - 2.4 (x^{2} + 2 h x + h^{2})$

Step 2.1.2.1.5

Apply the distributive property.

$f (x + h) = 2.9 + 5.8 x + 5.8 h - 2.4 x^{2} - 2.4 (2 h x) - 2.4 h^{2}$

Step 2.1.2.1.6

Multiply $2$ by $- 2.4$ .

$f (x + h) = 2.9 + 5.8 x + 5.8 h - 2.4 x^{2} - 4.8 h x - 2.4 h^{2}$

Step 2.1.2.2

The final answer is $2.9 + 5.8 x + 5.8 h - 2.4 x^{2} - 4.8 h x - 2.4 h^{2}$ .

$2.9 + 5.8 x + 5.8 h - 2.4 x^{2} - 4.8 h x - 2.4 h^{2}$

Step 2.2

Reorder.

Tap for more steps...

Step 2.2.1

Move $2.9$ .

$5.8 x + 5.8 h - 2.4 x^{2} - 4.8 h x - 2.4 h^{2} + 2.9$

Step 2.2.2

Move $5.8 x$ .

$5.8 h - 2.4 x^{2} - 4.8 h x - 2.4 h^{2} + 5.8 x + 2.9$

Step 2.2.3

Move $5.8 h$ .

$- 2.4 x^{2} - 4.8 h x - 2.4 h^{2} + 5.8 h + 5.8 x + 2.9$

Step 2.2.4

Move $- 2.4 x^{2}$ .

$- 4.8 h x - 2.4 h^{2} - 2.4 x^{2} + 5.8 h + 5.8 x + 2.9$

Step 2.2.5

Reorder $- 4.8 h x$ and $- 2.4 h^{2}$ .

$- 2.4 h^{2} - 4.8 h x - 2.4 x^{2} + 5.8 h + 5.8 x + 2.9$

Step 2.3

Find the components of the definition.

$f (x + h) = - 2.4 h^{2} - 4.8 h x - 2.4 x^{2} + 5.8 h + 5.8 x + 2.9$

$f (x) = 2.9 + 5.8 x - 2.4 x^{2}$

$f (x + h) = - 2.4 h^{2} - 4.8 h x - 2.4 x^{2} + 5.8 h + 5.8 x + 2.9$

$f (x) = 2.9 + 5.8 x - 2.4 x^{2}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 2.4 h^{2} - 4.8 h x - 2.4 x^{2} + 5.8 h + 5.8 x + 2.9 - (2.9 + 5.8 x - 2.4 x^{2})}{h}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Factor $0.1$ out of $- 2.4 h^{2} - 4.8 h x - 2.4 x^{2} + 5.8 h + 5.8 x + 2.9 - (2.9 + 5.8 x - 2.4 x^{2})$ .

Tap for more steps...

Step 4.1.1.1

Reorder the expression.

Tap for more steps...

Step 4.1.1.1.1

Move $h$ .

$\frac{- 2.4 h^{2} - 4.8 x h - 2.4 x^{2} + 5.8 h + 5.8 x + 2.9 - (2.9 + 5.8 x - 2.4 x^{2})}{h}$

Step 4.1.1.1.2

Move $2.9$ .

$\frac{- 2.4 h^{2} - 4.8 x h - 2.4 x^{2} + 5.8 h + 5.8 x + 2.9 - (5.8 x - 2.4 x^{2} + 2.9)}{h}$

Step 4.1.1.1.3

Reorder $5.8 x$ and $- 2.4 x^{2}$ .

$\frac{- 2.4 h^{2} - 4.8 x h - 2.4 x^{2} + 5.8 h + 5.8 x + 2.9 - (- 2.4 x^{2} + 5.8 x + 2.9)}{h}$

Step 4.1.1.2

Factor $0.1$ out of $- 2.4 h^{2}$ .

$\frac{0.1 (- 24 h^{2}) - 4.8 x h - 2.4 x^{2} + 5.8 h + 5.8 x + 2.9 - (- 2.4 x^{2} + 5.8 x + 2.9)}{h}$

Step 4.1.1.3

Factor $0.1$ out of $- 4.8 x h$ .

$\frac{0.1 (- 24 h^{2}) + 0.1 (- 48 x h) - 2.4 x^{2} + 5.8 h + 5.8 x + 2.9 - (- 2.4 x^{2} + 5.8 x + 2.9)}{h}$

Step 4.1.1.4

Factor $0.1$ out of $- 2.4 x^{2}$ .

$\frac{0.1 (- 24 h^{2}) + 0.1 (- 48 x h) + 0.1 (- 24 x^{2}) + 5.8 h + 5.8 x + 2.9 - (- 2.4 x^{2} + 5.8 x + 2.9)}{h}$

Step 4.1.1.5

Factor $0.1$ out of $5.8 h$ .

$\frac{0.1 (- 24 h^{2}) + 0.1 (- 48 x h) + 0.1 (- 24 x^{2}) + 0.1 (58 h) + 5.8 x + 2.9 - (- 2.4 x^{2} + 5.8 x + 2.9)}{h}$

Step 4.1.1.6

Factor $0.1$ out of $5.8 x$ .

$\frac{0.1 (- 24 h^{2}) + 0.1 (- 48 x h) + 0.1 (- 24 x^{2}) + 0.1 (58 h) + 0.1 (58 x) + 2.9 - (- 2.4 x^{2} + 5.8 x + 2.9)}{h}$

Step 4.1.1.7

Factor $0.1$ out of $2.9$ .

$\frac{0.1 (- 24 h^{2}) + 0.1 (- 48 x h) + 0.1 (- 24 x^{2}) + 0.1 (58 h) + 0.1 (58 x) + 0.1 (29) - (- 2.4 x^{2} + 5.8 x + 2.9)}{h}$

Step 4.1.1.8

Factor $0.1$ out of $- (- 2.4 x^{2} + 5.8 x + 2.9)$ .

$\frac{0.1 (- 24 h^{2}) + 0.1 (- 48 x h) + 0.1 (- 24 x^{2}) + 0.1 (58 h) + 0.1 (58 x) + 0.1 (29) + 0.1 (- 10 (- 2.4 x^{2} + 5.8 x + 2.9))}{h}$

Step 4.1.1.9

Factor $0.1$ out of $0.1 (- 24 h^{2}) + 0.1 (- 48 x h)$ .

$\frac{0.1 (- 24 h^{2} - 48 x h) + 0.1 (- 24 x^{2}) + 0.1 (58 h) + 0.1 (58 x) + 0.1 (29) + 0.1 (- 10 (- 2.4 x^{2} + 5.8 x + 2.9))}{h}$

Step 4.1.1.10

Factor $0.1$ out of $0.1 (- 24 h^{2} - 48 x h) + 0.1 (- 24 x^{2})$ .

$\frac{0.1 (- 24 h^{2} - 48 x h - 24 x^{2}) + 0.1 (58 h) + 0.1 (58 x) + 0.1 (29) + 0.1 (- 10 (- 2.4 x^{2} + 5.8 x + 2.9))}{h}$

Step 4.1.1.11

Factor $0.1$ out of $0.1 (- 24 h^{2} - 48 x h - 24 x^{2}) + 0.1 (58 h)$ .

$\frac{0.1 (- 24 h^{2} - 48 x h - 24 x^{2} + 58 h) + 0.1 (58 x) + 0.1 (29) + 0.1 (- 10 (- 2.4 x^{2} + 5.8 x + 2.9))}{h}$

Step 4.1.1.12

Factor $0.1$ out of $0.1 (- 24 h^{2} - 48 x h - 24 x^{2} + 58 h) + 0.1 (58 x)$ .

$\frac{0.1 (- 24 h^{2} - 48 x h - 24 x^{2} + 58 h + 58 x) + 0.1 (29) + 0.1 (- 10 (- 2.4 x^{2} + 5.8 x + 2.9))}{h}$

Step 4.1.1.13

Factor $0.1$ out of $0.1 (- 24 h^{2} - 48 x h - 24 x^{2} + 58 h + 58 x) + 0.1 (29)$ .

$\frac{0.1 (- 24 h^{2} - 48 x h - 24 x^{2} + 58 h + 58 x + 29) + 0.1 (- 10 (- 2.4 x^{2} + 5.8 x + 2.9))}{h}$

Step 4.1.1.14

Factor $0.1$ out of $0.1 (- 24 h^{2} - 48 x h - 24 x^{2} + 58 h + 58 x + 29) + 0.1 (- 10 (- 2.4 x^{2} + 5.8 x + 2.9))$ .

$\frac{0.1 (- 24 h^{2} - 48 x h - 24 x^{2} + 58 h + 58 x + 29 - 10 (- 2.4 x^{2} + 5.8 x + 2.9))}{h}$

Step 4.1.2

Apply the distributive property.

$\frac{0.1 (- 24 h^{2} - 48 x h - 24 x^{2} + 58 h + 58 x + 29 - 10 (- 2.4 x^{2}) - 10 (5.8 x) - 10 \cdot 2.9)}{h}$

Step 4.1.3

Simplify.

Tap for more steps...

Step 4.1.3.1

Multiply $- 2.4$ by $- 10$ .

$\frac{0.1 (- 24 h^{2} - 48 x h - 24 x^{2} + 58 h + 58 x + 29 + 24 x^{2} - 10 (5.8 x) - 10 \cdot 2.9)}{h}$

Step 4.1.3.2

Multiply $5.8$ by $- 10$ .

$\frac{0.1 (- 24 h^{2} - 48 x h - 24 x^{2} + 58 h + 58 x + 29 + 24 x^{2} - 58 x - 10 \cdot 2.9)}{h}$

Step 4.1.3.3

Multiply $- 10$ by $2.9$ .

$\frac{0.1 (- 24 h^{2} - 48 x h - 24 x^{2} + 58 h + 58 x + 29 + 24 x^{2} - 58 x - 29)}{h}$

Step 4.1.4

Add $- 24 x^{2}$ and $24 x^{2}$ .

$\frac{0.1 (- 24 h^{2} - 48 x h + 0 x^{2} + 58 h + 58 x + 29 - 58 x - 29)}{h}$

Step 4.1.5

Subtract $58 x$ from $58 x$ .

$\frac{0.1 (- 24 h^{2} - 48 x h + 0 x^{2} + 58 h + 0 x + 29 - 29)}{h}$

Step 4.1.6

Subtract $29$ from $29$ .

$\frac{0.1 (- 24 h^{2} - 48 x h + 0 x^{2} + 58 h + 0 x + 0)}{h}$

Step 4.1.7

Add $- 24 h^{2} - 48 x h + 0 x^{2} + 58 h + 0 x$ and $0$ .

$\frac{0.1 (- 24 h^{2} - 48 x h + 0 x^{2} + 58 h + 0 x)}{h}$

Step 4.1.8

Rewrite $- 24 h^{2} - 48 x h + 0 x^{2} + 58 h + 0 x$ in a factored form.

Tap for more steps...

Step 4.1.8.1

Factor $2$ out of $- 24 h^{2} - 48 x h + 0 x^{2} + 58 h + 0 x$ .

Tap for more steps...

Step 4.1.8.1.1

Factor $2$ out of $- 24 h^{2}$ .

$\frac{0.1 (2 (- 12 h^{2}) - 48 x h + 0 x^{2} + 58 h + 0 x)}{h}$

Step 4.1.8.1.2

Factor $2$ out of $- 48 x h$ .

$\frac{0.1 (2 (- 12 h^{2}) + 2 (- 24 x h) + 0 x^{2} + 58 h + 0 x)}{h}$

Step 4.1.8.1.3

Factor $2$ out of $0 x^{2}$ .

$\frac{0.1 (2 (- 12 h^{2}) + 2 (- 24 x h) + 2 (0 x^{2}) + 58 h + 0 x)}{h}$

Step 4.1.8.1.4

Factor $2$ out of $58 h$ .

$\frac{0.1 (2 (- 12 h^{2}) + 2 (- 24 x h) + 2 (0 x^{2}) + 2 (29 h) + 0 x)}{h}$

Step 4.1.8.1.5

Factor $2$ out of $0 x$ .

$\frac{0.1 (2 (- 12 h^{2}) + 2 (- 24 x h) + 2 (0 x^{2}) + 2 (29 h) + 2 (0 x))}{h}$

Step 4.1.8.1.6

Factor $2$ out of $2 (- 12 h^{2}) + 2 (- 24 x h)$ .

$\frac{0.1 (2 (- 12 h^{2} - 24 x h) + 2 (0 x^{2}) + 2 (29 h) + 2 (0 x))}{h}$

Step 4.1.8.1.7

Factor $2$ out of $2 (- 12 h^{2} - 24 x h) + 2 (0 x^{2})$ .

$\frac{0.1 (2 (- 12 h^{2} - 24 x h + 0 x^{2}) + 2 (29 h) + 2 (0 x))}{h}$

Step 4.1.8.1.8

Factor $2$ out of $2 (- 12 h^{2} - 24 x h + 0 x^{2}) + 2 (29 h)$ .

$\frac{0.1 (2 (- 12 h^{2} - 24 x h + 0 x^{2} + 29 h) + 2 (0 x))}{h}$

Step 4.1.8.1.9

Factor $2$ out of $2 (- 12 h^{2} - 24 x h + 0 x^{2} + 29 h) + 2 (0 x)$ .

$\frac{0.1 (2 (- 12 h^{2} - 24 x h + 0 x^{2} + 29 h + 0 x))}{h}$

Step 4.1.8.2

Multiply $0$ by $x^{2}$ .

$\frac{0.1 (2 (- 12 h^{2} - 24 x h + 0 + 29 h + 0 x))}{h}$

Step 4.1.8.3

Multiply $0$ by $x$ .

$\frac{0.1 (2 (- 12 h^{2} - 24 x h + 0 + 29 h + 0))}{h}$

Step 4.1.8.4

Add $- 12 h^{2} - 24 x h + 0 + 29 h$ and $0$ .

$\frac{0.1 (2 (- 12 h^{2} - 24 x h + 0 + 29 h))}{h}$

Step 4.1.8.5

Add $- 12 h^{2}$ and $0$ .

$\frac{0.1 (2 (- 12 h^{2} - 24 x h + 29 h))}{h}$

Step 4.1.8.6

Factor $h$ out of $- 12 h^{2} - 24 x h + 29 h$ .

Tap for more steps...

Step 4.1.8.6.1

Factor $h$ out of $- 12 h^{2}$ .

$\frac{0.1 (2 (h (- 12 h) - 24 x h + 29 h))}{h}$

Step 4.1.8.6.2

Factor $h$ out of $- 24 x h$ .

$\frac{0.1 (2 (h (- 12 h) + h (- 24 x) + 29 h))}{h}$

Step 4.1.8.6.3

Factor $h$ out of $29 h$ .

$\frac{0.1 (2 (h (- 12 h) + h (- 24 x) + h \cdot 29))}{h}$

Step 4.1.8.6.4

Factor $h$ out of $h (- 12 h) + h (- 24 x)$ .

$\frac{0.1 (2 (h (- 12 h - 24 x) + h \cdot 29))}{h}$

Step 4.1.8.6.5

Factor $h$ out of $h (- 12 h - 24 x) + h \cdot 29$ .

$\frac{0.1 (2 (h (- 12 h - 24 x + 29)))}{h}$

$\frac{0.1 (2 h (- 12 h - 24 x + 29))}{h}$

$\frac{0.1 \cdot 2 h (- 12 h - 24 x + 29)}{h}$

Step 4.1.9

Multiply $0.1$ by $2$ .

$\frac{0.2 h (- 12 h - 24 x + 29)}{h}$

Step 4.2

Simplify terms.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $h$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{0.2 h (- 12 h - 24 x + 29)}{h}$

Step 4.2.1.2

Divide $0.2 (- 12 h - 24 x + 29)$ by $1$ .

$0.2 (- 12 h - 24 x + 29)$

Step 4.2.2

Apply the distributive property.

$0.2 (- 12 h) + 0.2 (- 24 x) + 0.2 \cdot 29$

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

Multiply $- 12$ by $0.2$ .

$- 2.4 h + 0.2 (- 24 x) + 0.2 \cdot 29$

Step 4.3.2

Multiply $- 24$ by $0.2$ .

$- 2.4 h - 4.8 x + 0.2 \cdot 29$

Step 4.3.3

Multiply $0.2$ by $29$ .

$- 2.4 h - 4.8 x + 5.8$

Step 4.4

Reorder $- 2.4 h$ and $- 4.8 x$ .

$- 4.8 x - 2.4 h + 5.8$

Step 5