Algebra Examples

$4 x^{2} + 9 y^{2} - 16 x - 18 y - 11 = 0$

Move $- 11$ to the right side of the equation because it does not contain a variable.

$4 x^{2} + 9 y^{2} - 16 x - 18 y = 11$

Complete the square for $4 x^{2} - 16 x$ .

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

$a = 4, b = - 16, c = 0$

Consider the vertex form of a parabola.

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

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

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Multiply $2$ by $4$ to get $2 \cdot 4$ .

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

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $2 \cdot 4$ .

$d = - \frac{2 \cdot 8}{2 (4)}$

Cancel the common factor.

$d = - \frac{2 \cdot 8}{2 \cdot 4}$

Rewrite the expression.

$d = - \frac{8}{4}$

Simplify the expression.

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Divide $8$ by $4$ to get $2$ .

$d = - 1 \cdot 2$

Multiply $- 1$ by $2$ to get $- 2$ .

$d = - 2$

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

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Simplify each term.

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Multiply $4$ by $4$ to get $4 \cdot 4$ .

$e = 0 - \frac{{(- 16)}^{2}}{4 \cdot 4}$

Raise $- 16$ to the power of $2$ to get $256$ .

$e = 0 - \frac{256}{4 \cdot 4}$

Multiply $4$ by $4$ to get $16$ .

$e = 0 - \frac{256}{16}$

Divide $256$ by $16$ to get $16$ .

$e = 0 - 1 \cdot 16$

Multiply $- 1$ by $16$ to get $- 16$ .

$e = 0 - 16$

Subtract $16$ from $0$ to get $- 16$ .

$e = - 16$

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

$4 {(x - 2)}^{2} - 16$

Substitute $4 {(x - 2)}^{2} - 16$ for $4 x^{2} - 16 x$ in the equation $4 x^{2} + 9 y^{2} - 16 x - 18 y = 11$ .

$4 {(x - 2)}^{2} - 16 + 9 y^{2} - 18 y = 11$

Move $- 16$ to the right side of the equation by adding $16$ to both sides.

$4 {(x - 2)}^{2} + 9 y^{2} - 18 y = 11 + 16$

Complete the square for $9 y^{2} - 18 y$ .

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

$a = 9, b = - 18, c = 0$

Consider the vertex form of a parabola.

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

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

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Multiply $2$ by $9$ to get $2 \cdot 9$ .

$d = - \frac{18}{2 \cdot 9}$

Reduce the expression by cancelling the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

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Divide $9$ by $9$ to get $1$ .

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ to get $- 1$ .

$d = - 1$

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

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Simplify each term.

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Multiply $4$ by $9$ to get $4 \cdot 9$ .

$e = 0 - \frac{{(- 18)}^{2}}{4 \cdot 9}$

Raise $- 18$ to the power of $2$ to get $324$ .

$e = 0 - \frac{324}{4 \cdot 9}$

Multiply $4$ by $9$ to get $36$ .

$e = 0 - \frac{324}{36}$

Divide $324$ by $36$ to get $9$ .

$e = 0 - 1 \cdot 9$

Multiply $- 1$ by $9$ to get $- 9$ .

$e = 0 - 9$

Subtract $9$ from $0$ to get $- 9$ .

$e = - 9$

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

$9 {(y - 1)}^{2} - 9$

Substitute $9 {(y - 1)}^{2} - 9$ for $9 y^{2} - 18 y$ in the equation $4 x^{2} + 9 y^{2} - 16 x - 18 y = 11$ .

$4 {(x - 2)}^{2} + 9 {(y - 1)}^{2} - 9 = 11 + 16$

Move $- 9$ to the right side of the equation by adding $9$ to both sides.

$4 {(x - 2)}^{2} + 9 {(y - 1)}^{2} = 11 + 16 + 9$

Add $11$ and $16$ to get $27$ .

$4 {(x - 2)}^{2} + 9 {(y - 1)}^{2} = 27 + 9$

Add $27$ and $9$ to get $36$ .

$4 {(x - 2)}^{2} + 9 {(y - 1)}^{2} = 36$

Divide each term by $36$ to make the right side equal to $1$ .

$\frac{4 {(x - 2)}^{2}}{36} + \frac{9 {(y - 1)}^{2}}{36} = \frac{36}{36}$

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 $1$ .

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

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