Precalculus Examples

$9 x^{2} + 25 y^{2} - 72 x - 300 y + 819 = 0$

Step 1

Subtract $819$ from both sides of the equation.

$9 x^{2} + 25 y^{2} - 72 x - 300 y = - 819$

Step 2

Complete the square for $9 x^{2} - 72 x$ .

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Step 2.1

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

$a = 9$

$b = - 72$

$c = 0$

Step 2.2

Consider the vertex form of a parabola.

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

Step 2.3

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

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Step 2.3.1

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

$d = \frac{- 72}{2 \cdot 9}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $- 72$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $- 72$ .

$d = \frac{2 \cdot - 36}{2 \cdot 9}$

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot - 36}{2 (9)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 36}{2 \cdot 9}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 36}{9}$

Step 2.3.2.2

Cancel the common factor of $- 36$ and $9$ .

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Step 2.3.2.2.1

Factor $9$ out of $- 36$ .

$d = \frac{9 \cdot - 4}{9}$

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 4}{9 (1)}$

Step 2.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 4}{9 \cdot 1}$

Step 2.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.2.4

Divide $- 4$ by $1$ .

$d = - 4$

Step 2.4

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

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Step 2.4.1

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

$e = 0 - \frac{{(- 72)}^{2}}{4 \cdot 9}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Raise $- 72$ to the power of $2$ .

$e = 0 - \frac{5184}{4 \cdot 9}$

Step 2.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{5184}{36}$

Step 2.4.2.1.3

Divide $5184$ by $36$ .

$e = 0 - 1 \cdot 144$

Step 2.4.2.1.4

Multiply $- 1$ by $144$ .

$e = 0 - 144$

Step 2.4.2.2

Subtract $144$ from $0$ .

$e = - 144$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 4)}^{2} - 144$ .

$9 {(x - 4)}^{2} - 144$

Step 3

Substitute $9 {(x - 4)}^{2} - 144$ for $9 x^{2} - 72 x$ in the equation $9 x^{2} + 25 y^{2} - 72 x - 300 y = - 819$ .

$9 {(x - 4)}^{2} - 144 + 25 y^{2} - 300 y = - 819$

Step 4

Move $- 144$ to the right side of the equation by adding $144$ to both sides.

$9 {(x - 4)}^{2} + 25 y^{2} - 300 y = - 819 + 144$

Step 5

Complete the square for $25 y^{2} - 300 y$ .

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Step 5.1

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

$a = 25$

$b = - 300$

$c = 0$

Step 5.2

Consider the vertex form of a parabola.

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

Step 5.3

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

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Step 5.3.1

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

$d = \frac{- 300}{2 \cdot 25}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $- 300$ and $2$ .

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Step 5.3.2.1.1

Factor $2$ out of $- 300$ .

$d = \frac{2 \cdot - 150}{2 \cdot 25}$

Step 5.3.2.1.2

Cancel the common factors.

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Step 5.3.2.1.2.1

Factor $2$ out of $2 \cdot 25$ .

$d = \frac{2 \cdot - 150}{2 (25)}$

Step 5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 150}{2 \cdot 25}$

Step 5.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 150}{25}$

Step 5.3.2.2

Cancel the common factor of $- 150$ and $25$ .

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Step 5.3.2.2.1

Factor $25$ out of $- 150$ .

$d = \frac{25 \cdot - 6}{25}$

Step 5.3.2.2.2

Cancel the common factors.

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Step 5.3.2.2.2.1

Factor $25$ out of $25$ .

$d = \frac{25 \cdot - 6}{25 (1)}$

Step 5.3.2.2.2.2

Cancel the common factor.

$d = \frac{25 \cdot - 6}{25 \cdot 1}$

Step 5.3.2.2.2.3

Rewrite the expression.

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

Step 5.3.2.2.2.4

Divide $- 6$ by $1$ .

$d = - 6$

Step 5.4

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

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Step 5.4.1

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

$e = 0 - \frac{{(- 300)}^{2}}{4 \cdot 25}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

Raise $- 300$ to the power of $2$ .

$e = 0 - \frac{90000}{4 \cdot 25}$

Step 5.4.2.1.2

Multiply $4$ by $25$ .

$e = 0 - \frac{90000}{100}$

Step 5.4.2.1.3

Divide $90000$ by $100$ .

$e = 0 - 1 \cdot 900$

Step 5.4.2.1.4

Multiply $- 1$ by $900$ .

$e = 0 - 900$

Step 5.4.2.2

Subtract $900$ from $0$ .

$e = - 900$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(y - 6)}^{2} - 900$ .

$25 {(y - 6)}^{2} - 900$

Step 6

Substitute $25 {(y - 6)}^{2} - 900$ for $25 y^{2} - 300 y$ in the equation $9 x^{2} + 25 y^{2} - 72 x - 300 y = - 819$ .

$9 {(x - 4)}^{2} + 25 {(y - 6)}^{2} - 900 = - 819 + 144$

Step 7

Move $- 900$ to the right side of the equation by adding $900$ to both sides.

$9 {(x - 4)}^{2} + 25 {(y - 6)}^{2} = - 819 + 144 + 900$

Step 8

Simplify $- 819 + 144 + 900$ .

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Step 8.1

Add $- 819$ and $144$ .

$9 {(x - 4)}^{2} + 25 {(y - 6)}^{2} = - 675 + 900$

Step 8.2

Add $- 675$ and $900$ .

$9 {(x - 4)}^{2} + 25 {(y - 6)}^{2} = 225$

Step 9

Divide each term by $225$ to make the right side equal to one.

$\frac{9 {(x - 4)}^{2}}{225} + \frac{25 {(y - 6)}^{2}}{225} = \frac{225}{225}$

Step 10

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{{(x - 4)}^{2}}{25} + \frac{{(y - 6)}^{2}}{9} = 1$