Precalculus Examples

f (x) = x^{2} + 4 x - 5

$f (x) = x^{2} + 4 x - 5$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

x^{2} + 4 x - 5

$x^{2} + 4 x - 5$ .

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Step 1.1.1

Use the form

a x^{2} + b x + c

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

a

$a$ ,

b

$b$ , and

c

$c$ .

a = 1

$a = 1$

b = 4

$b = 4$

c = - 5

$c = - 5$

Step 1.1.2

Consider the vertex form of a parabola.

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

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

Step 1.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

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Step 1.1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

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

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

Step 1.1.3.2

Cancel the common factor of

4

$4$ and

2

$2$ .

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Step 1.1.3.2.1

Factor

2

$2$ out of

4

$4$ .

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

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

Step 1.1.3.2.2

Cancel the common factors.

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Step 1.1.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

d = \frac{2 \cdot 2}{2 (1)}

$d = \frac{2 \cdot 2}{2 (1)}$

Step 1.1.3.2.2.2

Cancel the common factor.

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

Step 1.1.3.2.2.3

Rewrite the expression.

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

Step 1.1.3.2.2.4

Divide

$2$ by

$1$ .

$d = 2$

$d = 2$

$d = 2$

$d = 2$

Step 1.1.4

Find the value of

$e$ using the formula

$e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.4.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

$e = c - \frac{b^{2}}{4 a}$ .

$e = - 5 - \frac{4^{2}}{4 \cdot 1}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Cancel the common factor of

$4^{2}$ and

$4$ .

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Step 1.1.4.2.1.1.1

Factor

$4$ out of

$4^{2}$ .

$e = - 5 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 1.1.4.2.1.1.2

Cancel the common factors.

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Step 1.1.4.2.1.1.2.1

Factor

$4$ out of

$4 \cdot 1$ .

$e = - 5 - \frac{4 \cdot 4}{4 (1)}$

Step 1.1.4.2.1.1.2.2

Cancel the common factor.

$e = - 5 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 1.1.4.2.1.1.2.3

Rewrite the expression.

$e = - 5 - \frac{4}{1}$

Step 1.1.4.2.1.1.2.4

Divide

$4$ by

$1$ .

$e = - 5 - 1 \cdot 4$

$e = - 5 - 1 \cdot 4$

$e = - 5 - 1 \cdot 4$

Step 1.1.4.2.1.2

Multiply

$- 1$ by

$4$ .

$e = - 5 - 4$

$e = - 5 - 4$

Step 1.1.4.2.2

Subtract

$4$ from

$- 5$ .

$e = - 9$

$e = - 9$

$e = - 9$

Step 1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

${(x + 2)}^{2} - 9$ .

${(x + 2)}^{2} - 9$

${(x + 2)}^{2} - 9$

Step 1.2

Set

$y$ equal to the new right side.

$y = {(x + 2)}^{2} - 9$

$y = {(x + 2)}^{2} - 9$

Step 2

Use the vertex form,

$y = a {(x - h)}^{2} + k$ , to determine the values of

$a$ ,

$h$ , and

$k$ .

$a = 1$

$h = - 2$

$k = - 9$

Step 3

Find the vertex

$(h, k)$ .

$(- 2, - 9)$

Step 4

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$