Precalculus Examples

$\frac{{(x - 17.5)}^{2}}{2.25} - \frac{{(y - 25)}^{2}}{2.25} = 1$

Step 1

Subtract $\frac{{(x - 17.5)}^{2}}{2.25}$ from both sides of the equation.

$- \frac{{(y - 25)}^{2}}{2.25} = 1 - \frac{{(x - 17.5)}^{2}}{2.25}$

Step 2

Simplify $- \frac{{(y - 25)}^{2}}{2.25}$ .

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Step 2.1

Multiply by $1$ .

$- \frac{1 {(y - 25)}^{2}}{2.25} = 1 - \frac{{(x - 17.5)}^{2}}{2.25}$

Step 2.2

Factor $2.25$ out of $2.25$ .

$- \frac{1 {(y - 25)}^{2}}{2.25 (1)} = 1 - \frac{{(x - 17.5)}^{2}}{2.25}$

Step 2.3

Separate fractions.

$- (\frac{1}{2.25} \cdot \frac{{(y - 25)}^{2}}{1}) = 1 - \frac{{(x - 17.5)}^{2}}{2.25}$

Step 2.4

Divide $1$ by $2.25$ .

$- (0.\overline{4} \frac{{(y - 25)}^{2}}{1}) = 1 - \frac{{(x - 17.5)}^{2}}{2.25}$

Step 2.5

Divide ${(y - 25)}^{2}$ by $1$ .

$- (0.\overline{4} {(y - 25)}^{2}) = 1 - \frac{{(x - 17.5)}^{2}}{2.25}$

Step 2.6

Multiply $0.\overline{4}$ by $- 1$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - \frac{{(x - 17.5)}^{2}}{2.25}$

Step 3

Simplify $1 - \frac{{(x - 17.5)}^{2}}{2.25}$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply by $1$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - \frac{1 {(x - 17.5)}^{2}}{2.25}$

Step 3.1.2

Factor $2.25$ out of $2.25$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - \frac{1 {(x - 17.5)}^{2}}{2.25 (1)}$

Step 3.1.3

Separate fractions.

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (\frac{1}{2.25} \cdot \frac{{(x - 17.5)}^{2}}{1})$

Step 3.1.4

Divide $1$ by $2.25$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} \frac{{(x - 17.5)}^{2}}{1})$

Step 3.1.5

Divide ${(x - 17.5)}^{2}$ by $1$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} {(x - 17.5)}^{2})$

Step 3.1.6

Rewrite ${(x - 17.5)}^{2}$ as $(x - 17.5) (x - 17.5)$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} ((x - 17.5) (x - 17.5)))$

Step 3.1.7

Expand $(x - 17.5) (x - 17.5)$ using the FOIL Method.

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Step 3.1.7.1

Apply the distributive property.

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} (x (x - 17.5) - 17.5 (x - 17.5)))$

Step 3.1.7.2

Apply the distributive property.

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} (x \cdot x + x \cdot - 17.5 - 17.5 (x - 17.5)))$

Step 3.1.7.3

Apply the distributive property.

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} (x \cdot x + x \cdot - 17.5 - 17.5 x - 17.5 \cdot - 17.5))$

Step 3.1.8

Simplify and combine like terms.

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Step 3.1.8.1

Simplify each term.

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Step 3.1.8.1.1

Multiply $x$ by $x$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} (x^{2} + x \cdot - 17.5 - 17.5 x - 17.5 \cdot - 17.5))$

Step 3.1.8.1.2

Move $- 17.5$ to the left of $x$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} (x^{2} - 17.5 \cdot x - 17.5 x - 17.5 \cdot - 17.5))$

Step 3.1.8.1.3

Multiply $- 17.5$ by $- 17.5$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} (x^{2} - 17.5 x - 17.5 x + 306.25))$

Step 3.1.8.2

Subtract $17.5 x$ from $- 17.5 x$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} (x^{2} - 35 x + 306.25))$

Step 3.1.9

Apply the distributive property.

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} x^{2} + 0.\overline{4} (- 35 x) + 0.\overline{4} \cdot 306.25)$

Step 3.1.10

Simplify.

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Step 3.1.10.1

Multiply $- 35$ by $0.\overline{4}$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} x^{2} - 15.\overline{5} x + 0.\overline{4} \cdot 306.25)$

Step 3.1.10.2

Multiply $0.\overline{4}$ by $306.25$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} x^{2} - 15.\overline{5} x + 136.\overline{1})$

Step 3.1.11

Apply the distributive property.

$- 0.\overline{4} {(y - 25)}^{2} = 1 - (0.\overline{4} x^{2}) - (- 15.\overline{5} x) - 1 \cdot 136.\overline{1}$

Step 3.1.12

Simplify.

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Step 3.1.12.1

Multiply $0.\overline{4}$ by $- 1$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - 0.\overline{4} x^{2} - (- 15.\overline{5} x) - 1 \cdot 136.\overline{1}$

Step 3.1.12.2

Multiply $- 15.\overline{5}$ by $- 1$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - 0.\overline{4} x^{2} + 15.\overline{5} x - 1 \cdot 136.\overline{1}$

Step 3.1.12.3

Multiply $- 1$ by $136.\overline{1}$ .

$- 0.\overline{4} {(y - 25)}^{2} = 1 - 0.\overline{4} x^{2} + 15.\overline{5} x - 136.\overline{1}$

Step 3.2

Subtract $136.\overline{1}$ from $1$ .

$- 0.\overline{4} {(y - 25)}^{2} = - 0.\overline{4} x^{2} + 15.\overline{5} x - 135.\overline{1}$

Step 4

Divide each term in $- 0.\overline{4} {(y - 25)}^{2} = - 0.\overline{4} x^{2} + 15.\overline{5} x - 135.\overline{1}$ by $- 0.\overline{4}$ and simplify.

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Step 4.1

Divide each term in $- 0.\overline{4} {(y - 25)}^{2} = - 0.\overline{4} x^{2} + 15.\overline{5} x - 135.\overline{1}$ by $- 0.\overline{4}$ .

$\frac{- 0.\overline{4} {(y - 25)}^{2}}{- 0.\overline{4}} = \frac{- 0.\overline{4} x^{2}}{- 0.\overline{4}} + \frac{15.\overline{5} x}{- 0.\overline{4}} + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 0.\overline{4}$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 0.\overline{4} {(y - 25)}^{2}}{- 0.\overline{4}} = \frac{- 0.\overline{4} x^{2}}{- 0.\overline{4}} + \frac{15.\overline{5} x}{- 0.\overline{4}} + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.2.1.2

Divide ${(y - 25)}^{2}$ by $1$ .

${(y - 25)}^{2} = \frac{- 0.\overline{4} x^{2}}{- 0.\overline{4}} + \frac{15.\overline{5} x}{- 0.\overline{4}} + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Cancel the common factor of $- 0.\overline{4}$ .

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Step 4.3.1.1.1

Cancel the common factor.

${(y - 25)}^{2} = \frac{- 0.\overline{4} x^{2}}{- 0.\overline{4}} + \frac{15.\overline{5} x}{- 0.\overline{4}} + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.3.1.1.2

Divide $x^{2}$ by $1$ .

${(y - 25)}^{2} = x^{2} + \frac{15.\overline{5} x}{- 0.\overline{4}} + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.3.1.2

Move the negative in front of the fraction.

${(y - 25)}^{2} = x^{2} - \frac{15.\overline{5} x}{0.\overline{4}} + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.3.1.3

Factor $15.\overline{5}$ out of $15.\overline{5} x$ .

${(y - 25)}^{2} = x^{2} - \frac{15.\overline{5} (x)}{0.\overline{4}} + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.3.1.4

Factor $0.\overline{4}$ out of $0.\overline{4}$ .

${(y - 25)}^{2} = x^{2} - \frac{15.\overline{5} (x)}{0.\overline{4} (1)} + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.3.1.5

Separate fractions.

${(y - 25)}^{2} = x^{2} - (\frac{15.\overline{5}}{0.\overline{4}} \cdot \frac{x}{1}) + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.3.1.6

Divide $15.\overline{5}$ by $0.\overline{4}$ .

${(y - 25)}^{2} = x^{2} - (35 \frac{x}{1}) + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.3.1.7

Divide $x$ by $1$ .

${(y - 25)}^{2} = x^{2} - (35 x) + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.3.1.8

Multiply $35$ by $- 1$ .

${(y - 25)}^{2} = x^{2} - 35 x + \frac{- 135.\overline{1}}{- 0.\overline{4}}$

Step 4.3.1.9

Divide $- 135.\overline{1}$ by $- 0.\overline{4}$ .

${(y - 25)}^{2} = x^{2} - 35 x + 304$

Step 5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y - 25 = \pm \sqrt{x^{2} - 35 x + 304}$

Step 6

Factor $x^{2} - 35 x + 304$ using the AC method.

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Step 6.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $304$ and whose sum is $- 35$ .

$- 19, - 16$

Step 6.2

Write the factored form using these integers.

$y - 25 = \pm \sqrt{(x - 19) (x - 16)}$

Step 7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.1

First, use the positive value of the $\pm$ to find the first solution.

$y - 25 = \sqrt{(x - 19) (x - 16)}$

Step 7.2

Add $25$ to both sides of the equation.

$y = \sqrt{(x - 19) (x - 16)} + 25$

Step 7.3

Next, use the negative value of the $\pm$ to find the second solution.

$y - 25 = - \sqrt{(x - 19) (x - 16)}$

Step 7.4

Add $25$ to both sides of the equation.

$y = - \sqrt{(x - 19) (x - 16)} + 25$

Step 7.5

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{(x - 19) (x - 16)} + 25$

$y = - \sqrt{(x - 19) (x - 16)} + 25$

$y = \sqrt{(x - 19) (x - 16)} + 25$

$y = - \sqrt{(x - 19) (x - 16)} + 25$

Step 8

To rewrite as a function of $x$ , write the equation so that $y$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = \sqrt{(x - 19) (x - 16)} + 25, - \sqrt{(x - 19) (x - 16)} + 25$