Calculus Examples

$3 - 2 x - x^{2}$

Step 1

Simplify the expression.

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Step 1.1

Move $3$ .

$- 2 x - x^{2} + 3$

Step 1.2

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

$- x^{2} - 2 x + 3$

Step 2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = - 2$

$c = 3$

Step 3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 2}{2 \cdot - 1}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Cancel the common factor of $- 2$ and $2$ .

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Step 4.2.1.1

Factor $2$ out of $- 2$ .

$d = \frac{2 \cdot - 1}{2 \cdot - 1}$

Step 4.2.1.2

Move the negative one from the denominator of $\frac{- 1}{- 1}$ .

$d = - 1 \cdot - 1$

Step 4.2.2

Rewrite $- 1 \cdot - 1$ as $- - 1$ .

$d = - - 1$

Step 4.2.3

Multiply $- 1$ by $- 1$ .

$d = 1$

Step 5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 3 - \frac{{(- 2)}^{2}}{4 \cdot - 1}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 2$ to the power of $2$ .

$e = 3 - \frac{4}{4 \cdot - 1}$

Step 5.2.1.2

Multiply $4$ by $- 1$ .

$e = 3 - \frac{4}{- 4}$

Step 5.2.1.3

Divide $4$ by $- 4$ .

$e = 3 - - 1$

Step 5.2.1.4

Multiply $- 1$ by $- 1$ .

$e = 3 + 1$

Step 5.2.2

Add $3$ and $1$ .

$e = 4$

Step 6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + 1)}^{2} + 4$ .

$- {(x + 1)}^{2} + 4$