Precalculus Examples

$f (x) = {(x + 1)}^{2} - \frac{25}{9}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $y$ to the left side of the equation.

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Step 1.1.1

To write ${(x + 1)}^{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = {(x + 1)}^{2} \cdot \frac{9}{9} - \frac{25}{9}$

Step 1.1.2

Combine ${(x + 1)}^{2}$ and $\frac{9}{9}$ .

$y = \frac{{(x + 1)}^{2} \cdot 9}{9} - \frac{25}{9}$

Step 1.1.3

Combine the numerators over the common denominator.

$y = \frac{{(x + 1)}^{2} \cdot 9 - 25}{9}$

Step 1.1.4

Reorder terms.

$y = \frac{1}{9} \cdot ({(x + 1)}^{2} \cdot 9 - 25)$

Step 1.2

Complete the square for $\frac{1}{9} \cdot ({(x + 1)}^{2} \cdot 9 - 25)$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$\frac{1}{9} \cdot ((x + 1) (x + 1) \cdot 9 - 25)$

Step 1.2.1.1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$\frac{1}{9} \cdot ((x (x + 1) + 1 (x + 1)) \cdot 9 - 25)$

Step 1.2.1.1.2.2

Apply the distributive property.

$\frac{1}{9} \cdot ((x \cdot x + x \cdot 1 + 1 (x + 1)) \cdot 9 - 25)$

Step 1.2.1.1.2.3

Apply the distributive property.

$\frac{1}{9} \cdot ((x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) \cdot 9 - 25)$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $x$ by $x$ .

$\frac{1}{9} \cdot ((x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) \cdot 9 - 25)$

Step 1.2.1.1.3.1.2

Multiply $x$ by $1$ .

$\frac{1}{9} \cdot ((x^{2} + x + 1 x + 1 \cdot 1) \cdot 9 - 25)$

Step 1.2.1.1.3.1.3

Multiply $x$ by $1$ .

$\frac{1}{9} \cdot ((x^{2} + x + x + 1 \cdot 1) \cdot 9 - 25)$

Step 1.2.1.1.3.1.4

Multiply $1$ by $1$ .

$\frac{1}{9} \cdot ((x^{2} + x + x + 1) \cdot 9 - 25)$

Step 1.2.1.1.3.2

Add $x$ and $x$ .

$\frac{1}{9} \cdot ((x^{2} + 2 x + 1) \cdot 9 - 25)$

Step 1.2.1.1.4

Apply the distributive property.

$\frac{1}{9} \cdot (x^{2} \cdot 9 + 2 x \cdot 9 + 1 \cdot 9 - 25)$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Move $9$ to the left of $x^{2}$ .

$\frac{1}{9} \cdot (9 \cdot x^{2} + 2 x \cdot 9 + 1 \cdot 9 - 25)$

Step 1.2.1.1.5.2

Multiply $9$ by $2$ .

$\frac{1}{9} \cdot (9 \cdot x^{2} + 18 x + 1 \cdot 9 - 25)$

Step 1.2.1.1.5.3

Multiply $9$ by $1$ .

$\frac{1}{9} \cdot (9 \cdot x^{2} + 18 x + 9 - 25)$

$\frac{1}{9} \cdot (9 x^{2} + 18 x + 9 - 25)$

Step 1.2.1.2

Subtract $25$ from $9$ .

$\frac{1}{9} \cdot (9 x^{2} + 18 x - 16)$

Step 1.2.1.3

Apply the distributive property.

$\frac{1}{9} (9 x^{2}) + \frac{1}{9} (18 x) + \frac{1}{9} \cdot - 16$

Step 1.2.1.4

Simplify.

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Step 1.2.1.4.1

Cancel the common factor of $9$ .

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Step 1.2.1.4.1.1

Factor $9$ out of $9 x^{2}$ .

$\frac{1}{9} (9 (x^{2})) + \frac{1}{9} (18 x) + \frac{1}{9} \cdot - 16$

Step 1.2.1.4.1.2

Cancel the common factor.

$\frac{1}{9} (9 x^{2}) + \frac{1}{9} (18 x) + \frac{1}{9} \cdot - 16$

Step 1.2.1.4.1.3

Rewrite the expression.

$x^{2} + \frac{1}{9} (18 x) + \frac{1}{9} \cdot - 16$

Step 1.2.1.4.2

Cancel the common factor of $9$ .

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Step 1.2.1.4.2.1

Factor $9$ out of $18 x$ .

$x^{2} + \frac{1}{9} (9 (2 x)) + \frac{1}{9} \cdot - 16$

Step 1.2.1.4.2.2

Cancel the common factor.

$x^{2} + \frac{1}{9} (9 (2 x)) + \frac{1}{9} \cdot - 16$

Step 1.2.1.4.2.3

Rewrite the expression.

$x^{2} + 2 x + \frac{1}{9} \cdot - 16$

Step 1.2.1.4.3

Combine $\frac{1}{9}$ and $- 16$ .

$x^{2} + 2 x + \frac{- 16}{9}$

Step 1.2.1.5

Move the negative in front of the fraction.

$x^{2} + 2 x - \frac{16}{9}$

Step 1.2.2

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

$a = 1$

$b = 2$

$c = - \frac{16}{9}$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.1

Cancel the common factor.

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

Step 1.2.4.2.2

Rewrite the expression.

$d = 1$

Step 1.2.5

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

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Step 1.2.5.1

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

$e = - \frac{16}{9} - \frac{2^{2}}{4 \cdot 1}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

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

$e = - \frac{16}{9} - \frac{4}{4 \cdot 1}$

Step 1.2.5.2.1.2

Multiply $4$ by $1$ .

$e = - \frac{16}{9} - \frac{4}{4}$

Step 1.2.5.2.1.3

Cancel the common factor of $4$ .

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Step 1.2.5.2.1.3.1

Cancel the common factor.

$e = - \frac{16}{9} - \frac{4}{4}$

Step 1.2.5.2.1.3.2

Rewrite the expression.

$e = - \frac{16}{9} - 1 \cdot 1$

Step 1.2.5.2.1.4

Multiply $- 1$ by $1$ .

$e = - \frac{16}{9} - 1$

Step 1.2.5.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$e = - \frac{16}{9} - 1 \cdot \frac{9}{9}$

Step 1.2.5.2.3

Combine $- 1$ and $\frac{9}{9}$ .

$e = - \frac{16}{9} + \frac{- 1 \cdot 9}{9}$

Step 1.2.5.2.4

Combine the numerators over the common denominator.

$e = \frac{- 16 - 1 \cdot 9}{9}$

Step 1.2.5.2.5

Simplify the numerator.

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Step 1.2.5.2.5.1

Multiply $- 1$ by $9$ .

$e = \frac{- 16 - 9}{9}$

Step 1.2.5.2.5.2

Subtract $9$ from $- 16$ .

$e = \frac{- 25}{9}$

Step 1.2.5.2.6

Move the negative in front of the fraction.

$e = - \frac{25}{9}$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 1)}^{2} - \frac{25}{9}$ .

${(x + 1)}^{2} - \frac{25}{9}$

Step 1.3

Set $y$ equal to the new right side.

$y = {(x + 1)}^{2} - \frac{25}{9}$

Step 2

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

$a = 1$

$h = - 1$

$k = - \frac{25}{9}$

Step 3

Find the vertex $(h, k)$ .

$(- 1, - \frac{25}{9})$

Step 4