Algebra Examples

$9 x^{2} + y^{2} + 144 x = - 567$

Step 1

Complete the square for $9 x^{2} + 144 x$ .

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Step 1.1

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

$a = 9$

$b = 144$

$c = 0$

Step 1.2

Consider the vertex form of a parabola.

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

Step 1.3

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

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Step 1.3.1

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

$d = \frac{144}{2 \cdot 9}$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Cancel the common factor of $144$ and $2$ .

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Step 1.3.2.1.1

Factor $2$ out of $144$ .

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

Step 1.3.2.1.2

Cancel the common factors.

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Step 1.3.2.1.2.1

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

$d = \frac{2 \cdot 72}{2 (9)}$

Step 1.3.2.1.2.2

Cancel the common factor.

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

Step 1.3.2.1.2.3

Rewrite the expression.

$d = \frac{72}{9}$

Step 1.3.2.2

Cancel the common factor of $72$ and $9$ .

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Step 1.3.2.2.1

Factor $9$ out of $72$ .

$d = \frac{9 \cdot 8}{9}$

Step 1.3.2.2.2

Cancel the common factors.

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Step 1.3.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot 8}{9 (1)}$

Step 1.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot 8}{9 \cdot 1}$

Step 1.3.2.2.2.3

Rewrite the expression.

$d = \frac{8}{1}$

Step 1.3.2.2.2.4

Divide $8$ by $1$ .

$d = 8$

Step 1.4

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

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Step 1.4.1

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

$e = 0 - \frac{144^{2}}{4 \cdot 9}$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Raise $144$ to the power of $2$ .

$e = 0 - \frac{20736}{4 \cdot 9}$

Step 1.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{20736}{36}$

Step 1.4.2.1.3

Divide $20736$ by $36$ .

$e = 0 - 1 \cdot 576$

Step 1.4.2.1.4

Multiply $- 1$ by $576$ .

$e = 0 - 576$

Step 1.4.2.2

Subtract $576$ from $0$ .

$e = - 576$

Step 1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x + 8)}^{2} - 576$ .

$9 {(x + 8)}^{2} - 576$

Step 2

Substitute $9 {(x + 8)}^{2} - 576$ for $9 x^{2} + 144 x$ in the equation $9 x^{2} + y^{2} + 144 x = - 567$ .

$9 {(x + 8)}^{2} - 576 + y^{2} = - 567$

Step 3

Move $- 576$ to the right side of the equation by adding $576$ to both sides.

$9 {(x + 8)}^{2} + y^{2} = - 567 + 576$

Step 4

Add $- 567$ and $576$ .

$9 {(x + 8)}^{2} + y^{2} = 9$

Step 5

Divide each term by $9$ to make the right side equal to one.

$\frac{9 {(x + 8)}^{2}}{9} + \frac{y^{2}}{9} = \frac{9}{9}$

Step 6

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 $1$ .

${(x + 8)}^{2} + \frac{y^{2}}{9} = 1$