Precalculus Examples

$25 x^{2} - 4 y^{2} + 300 x + 32 y + 936 = 0$

Step 1

Subtract $936$ from both sides of the equation.

$25 x^{2} - 4 y^{2} + 300 x + 32 y = - 936$

Step 2

Complete the square for $25 x^{2} + 300 x$ .

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Step 2.1

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

$a = 25$

$b = 300$

$c = 0$

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

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Factor $2$ out of $300$ .

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

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 $2 \cdot 25$ .

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

Step 2.3.2.1.2.2

Cancel the common factor.

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

Step 2.3.2.1.2.3

Rewrite the expression.

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

Step 2.3.2.2

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

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Step 2.3.2.2.1

Factor $25$ out of $150$ .

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

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $25$ out of $25$ .

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

Step 2.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.2.4

Divide $6$ by $1$ .

$d = 6$

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

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

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 $300$ to the power of $2$ .

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

Step 2.4.2.1.2

Multiply $4$ by $25$ .

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

Step 2.4.2.1.3

Divide $90000$ by $100$ .

$e = 0 - 1 \cdot 900$

Step 2.4.2.1.4

Multiply $- 1$ by $900$ .

$e = 0 - 900$

Step 2.4.2.2

Subtract $900$ from $0$ .

$e = - 900$

Step 2.5

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

$25 {(x + 6)}^{2} - 900$

Step 3

Substitute $25 {(x + 6)}^{2} - 900$ for $25 x^{2} + 300 x$ in the equation $25 x^{2} - 4 y^{2} + 300 x + 32 y = - 936$ .

$25 {(x + 6)}^{2} - 900 - 4 y^{2} + 32 y = - 936$

Step 4

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

$25 {(x + 6)}^{2} - 4 y^{2} + 32 y = - 936 + 900$

Step 5

Complete the square for $- 4 y^{2} + 32 y$ .

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Step 5.1

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

$a = - 4$

$b = 32$

$c = 0$

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

$d = \frac{32}{2 \cdot - 4}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $32$ and $2$ .

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Step 5.3.2.1.1

Factor $2$ out of $32$ .

$d = \frac{2 \cdot 16}{2 \cdot - 4}$

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 $2 \cdot - 4$ .

$d = \frac{2 \cdot 16}{2 (- 4)}$

Step 5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 16}{2 \cdot - 4}$

Step 5.3.2.1.2.3

Rewrite the expression.

$d = \frac{16}{- 4}$

Step 5.3.2.2

Cancel the common factor of $16$ and $- 4$ .

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Step 5.3.2.2.1

Factor $4$ out of $16$ .

$d = \frac{4 (4)}{- 4}$

Step 5.3.2.2.2

Move the negative one from the denominator of $\frac{4}{- 1}$ .

$d = - 1 \cdot 4$

Step 5.3.2.3

Multiply $- 1$ by $4$ .

$d = - 4$

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

$e = 0 - \frac{32^{2}}{4 \cdot - 4}$

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 $32$ to the power of $2$ .

$e = 0 - \frac{1024}{4 \cdot - 4}$

Step 5.4.2.1.2

Multiply $4$ by $- 4$ .

$e = 0 - \frac{1024}{- 16}$

Step 5.4.2.1.3

Divide $1024$ by $- 16$ .

$e = 0 - - 64$

Step 5.4.2.1.4

Multiply $- 1$ by $- 64$ .

$e = 0 + 64$

Step 5.4.2.2

Add $0$ and $64$ .

$e = 64$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(y - 4)}^{2} + 64$ .

$- 4 {(y - 4)}^{2} + 64$

Step 6

Substitute $- 4 {(y - 4)}^{2} + 64$ for $- 4 y^{2} + 32 y$ in the equation $25 x^{2} - 4 y^{2} + 300 x + 32 y = - 936$ .

$25 {(x + 6)}^{2} - 4 {(y - 4)}^{2} + 64 = - 936 + 900$

Step 7

Move $64$ to the right side of the equation by adding $64$ to both sides.

$25 {(x + 6)}^{2} - 4 {(y - 4)}^{2} = - 936 + 900 - 64$

Step 8

Simplify $- 936 + 900 - 64$ .

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Step 8.1

Add $- 936$ and $900$ .

$25 {(x + 6)}^{2} - 4 {(y - 4)}^{2} = - 36 - 64$

Step 8.2

Subtract $64$ from $- 36$ .

$25 {(x + 6)}^{2} - 4 {(y - 4)}^{2} = - 100$

Step 9

Flip the sign on each term of the equation so the term on the right side is positive.

$- 25 {(x + 6)}^{2} + 4 {(y - 4)}^{2} = 100$

Step 10

Divide each term by $100$ to make the right side equal to one.

$- \frac{25 {(x + 6)}^{2}}{100} + \frac{4 {(y - 4)}^{2}}{100} = \frac{100}{100}$

Step 11

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 $1$ .

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