Precalculus Examples

$- x^{2} + 4 x$

Step 1

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

$a = - 1$

$b = 4$

$c = 0$

Step 2

Consider the vertex form of a parabola.

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

Step 3

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

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Step 3.1

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

$d = \frac{4}{2 \cdot - 1}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Cancel the common factor of $4$ and $2$ .

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Step 3.2.1.1

Factor $2$ out of $4$ .

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

Step 3.2.1.2

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$d = - 1 \cdot 2$

Step 3.2.2

Multiply $- 1$ by $2$ .

$d = - 2$

Step 4

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

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Step 4.1

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

$e = 0 - \frac{4^{2}}{4 \cdot - 1}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Cancel the common factor of $4^{2}$ and $4$ .

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Step 4.2.1.1.1

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 \cdot - 1}$

Step 4.2.1.1.2

Move the negative one from the denominator of $\frac{4}{- 1}$ .

$e = 0 - (- 1 \cdot 4)$

Step 4.2.1.2

Multiply $- (- 1 \cdot 4)$ .

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Step 4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - - 4$

Step 4.2.1.2.2

Multiply $- 1$ by $- 4$ .

$e = 0 + 4$

Step 4.2.2

Add $0$ and $4$ .

$e = 4$

Step 5

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

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