Precalculus Examples

$f (x) = \frac{6}{5} \cdot (x^{2} + 2 x - 5)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Simplify $\frac{6}{5} \cdot (x^{2} + 2 x - 5)$ .

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Step 1.1.1

Apply the distributive property.

$y = \frac{6}{5} x^{2} + \frac{6}{5} (2 x) + \frac{6}{5} \cdot - 5$

Step 1.1.2

Simplify.

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Step 1.1.2.1

Combine $\frac{6}{5}$ and $x^{2}$ .

$y = \frac{6 x^{2}}{5} + \frac{6}{5} (2 x) + \frac{6}{5} \cdot - 5$

Step 1.1.2.2

Multiply $\frac{6}{5} (2 x)$ .

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Step 1.1.2.2.1

Combine $2$ and $\frac{6}{5}$ .

$y = \frac{6 x^{2}}{5} + \frac{2 \cdot 6}{5} x + \frac{6}{5} \cdot - 5$

Step 1.1.2.2.2

Multiply $2$ by $6$ .

$y = \frac{6 x^{2}}{5} + \frac{12}{5} x + \frac{6}{5} \cdot - 5$

Step 1.1.2.2.3

Combine $\frac{12}{5}$ and $x$ .

$y = \frac{6 x^{2}}{5} + \frac{12 x}{5} + \frac{6}{5} \cdot - 5$

Step 1.1.2.3

Cancel the common factor of $5$ .

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Step 1.1.2.3.1

Factor $5$ out of $- 5$ .

$y = \frac{6 x^{2}}{5} + \frac{12 x}{5} + \frac{6}{5} \cdot (5 (- 1))$

Step 1.1.2.3.2

Cancel the common factor.

$y = \frac{6 x^{2}}{5} + \frac{12 x}{5} + \frac{6}{5} \cdot (5 \cdot - 1)$

Step 1.1.2.3.3

Rewrite the expression.

$y = \frac{6 x^{2}}{5} + \frac{12 x}{5} + 6 \cdot - 1$

Step 1.1.2.4

Multiply $6$ by $- 1$ .

$y = \frac{6 x^{2}}{5} + \frac{12 x}{5} - 6$

Step 1.2

Complete the square for $\frac{6 x^{2}}{5} + \frac{12 x}{5} - 6$ .

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Step 1.2.1

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

$a = \frac{6}{5}$

$b = \frac{12}{5}$

$c = - 6$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

$d = \frac{\frac{12}{5}}{2 (\frac{6}{5})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{12}{5} \cdot \frac{1}{2 (\frac{6}{5})}$

Step 1.2.3.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.1

Factor $2$ out of $12$ .

$d = \frac{2 (6)}{5} \cdot \frac{1}{2 (\frac{6}{5})}$

Step 1.2.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 6}{5} \cdot \frac{1}{2 (\frac{6}{5})}$

Step 1.2.3.2.2.3

Rewrite the expression.

$d = \frac{6}{5} \cdot \frac{1}{\frac{6}{5}}$

Step 1.2.3.2.3

Multiply $\frac{6}{5}$ by $\frac{1}{\frac{6}{5}}$ .

$d = \frac{6}{5 (\frac{6}{5})}$

Step 1.2.3.2.4

Combine $5$ and $\frac{6}{5}$ .

$d = \frac{6}{\frac{5 \cdot 6}{5}}$

Step 1.2.3.2.5

Multiply $5$ by $6$ .

$d = \frac{6}{\frac{30}{5}}$

Step 1.2.3.2.6

Cancel the common factor of $30$ and $5$ .

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Step 1.2.3.2.6.1

Factor $5$ out of $30$ .

$d = \frac{6}{\frac{5 \cdot 6}{5}}$

Step 1.2.3.2.6.2

Cancel the common factors.

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Step 1.2.3.2.6.2.1

Factor $5$ out of $5$ .

$d = \frac{6}{\frac{5 \cdot 6}{5 (1)}}$

Step 1.2.3.2.6.2.2

Cancel the common factor.

$d = \frac{6}{\frac{5 \cdot 6}{5 \cdot 1}}$

Step 1.2.3.2.6.2.3

Rewrite the expression.

$d = \frac{6}{\frac{6}{1}}$

Step 1.2.3.2.6.2.4

Divide $6$ by $1$ .

$d = \frac{6}{6}$

Step 1.2.3.2.7

Cancel the common factor of $6$ .

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Step 1.2.3.2.7.1

Cancel the common factor.

$d = \frac{6}{6}$

Step 1.2.3.2.7.2

Rewrite the expression.

$d = 1$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = - 6 - \frac{{(\frac{12}{5})}^{2}}{4 (\frac{6}{5})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $\frac{12}{5}$ .

$e = - 6 - \frac{\frac{12^{2}}{5^{2}}}{4 (\frac{6}{5})}$

Step 1.2.4.2.1.1.2

Raise $12$ to the power of $2$ .

$e = - 6 - \frac{\frac{144}{5^{2}}}{4 (\frac{6}{5})}$

Step 1.2.4.2.1.1.3

Raise $5$ to the power of $2$ .

$e = - 6 - \frac{\frac{144}{25}}{4 (\frac{6}{5})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{6}{5}$ .

$e = - 6 - \frac{\frac{144}{25}}{\frac{4 \cdot 6}{5}}$

Step 1.2.4.2.1.3

Multiply $4$ by $6$ .

$e = - 6 - \frac{\frac{144}{25}}{\frac{24}{5}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - 6 - (\frac{144}{25} \cdot \frac{5}{24})$

Step 1.2.4.2.1.5

Cancel the common factor of $24$ .

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Step 1.2.4.2.1.5.1

Factor $24$ out of $144$ .

$e = - 6 - (\frac{24 (6)}{25} \cdot \frac{5}{24})$

Step 1.2.4.2.1.5.2

Cancel the common factor.

$e = - 6 - (\frac{24 \cdot 6}{25} \cdot \frac{5}{24})$

Step 1.2.4.2.1.5.3

Rewrite the expression.

$e = - 6 - (\frac{6}{25} \cdot 5)$

Step 1.2.4.2.1.6

Cancel the common factor of $5$ .

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Step 1.2.4.2.1.6.1

Factor $5$ out of $25$ .

$e = - 6 - (\frac{6}{5 (5)} \cdot 5)$

Step 1.2.4.2.1.6.2

Cancel the common factor.

$e = - 6 - (\frac{6}{5 \cdot 5} \cdot 5)$

Step 1.2.4.2.1.6.3

Rewrite the expression.

$e = - 6 - \frac{6}{5}$

Step 1.2.4.2.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$e = - 6 \cdot \frac{5}{5} - \frac{6}{5}$

Step 1.2.4.2.3

Combine $- 6$ and $\frac{5}{5}$ .

$e = \frac{- 6 \cdot 5}{5} - \frac{6}{5}$

Step 1.2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 6 \cdot 5 - 6}{5}$

Step 1.2.4.2.5

Simplify the numerator.

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Step 1.2.4.2.5.1

Multiply $- 6$ by $5$ .

$e = \frac{- 30 - 6}{5}$

Step 1.2.4.2.5.2

Subtract $6$ from $- 30$ .

$e = \frac{- 36}{5}$

Step 1.2.4.2.6

Move the negative in front of the fraction.

$e = - \frac{36}{5}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{6}{5} {(x + 1)}^{2} - \frac{36}{5}$ .

$\frac{6}{5} {(x + 1)}^{2} - \frac{36}{5}$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{6}{5} \cdot {(x + 1)}^{2} - \frac{36}{5}$

Step 2

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

$a = \frac{6}{5}$

$h = - 1$

$k = - \frac{36}{5}$

Step 3

Find the vertex $(h, k)$ .

$(- 1, - \frac{36}{5})$

Step 4