Algebra Examples

$4 x^{2} + y^{2} - 16 x + 2 y + 13 = 0$

Move $13$ to the right side of the equation because it does not contain a variable.

$4 x^{2} + y^{2} - 16 x + 2 y = - 13$

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 $4$ .

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

Reduce the expression by cancelling the common factors.

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Factor $2$ out of $16$ .

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

Cancel 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 $4$ .

$d = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$d = - 2$

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 $4$ .

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

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

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

Multiply $4$ by $4$ .

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

Divide $256$ by $16$ .

$e = 0 - 1 \cdot 16$

Multiply $- 1$ by $16$ .

$e = 0 - 16$

Subtract $16$ from $0$ .

$e = - 16$

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 $4 x^{2} + y^{2} - 16 x + 2 y = - 13$ .

$4 {(x - 2)}^{2} - 16 + y^{2} + 2 y = - 13$

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

$4 {(x - 2)}^{2} + y^{2} + 2 y = - 13 + 16$

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

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

$a = 1, b = 2, c = 0$

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 $1$ .

$d = \frac{2}{2 \cdot 1}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor.

$d = \frac{2}{2 \cdot 1}$

Divide $1$ by $1$ .

$d = 1$

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 $1$ .

$e = 0 - \frac{{(2)}^{2}}{4 \cdot 1}$

Simplify the numerator.

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Remove parentheses around $2$ .

$e = 0 - \frac{2^{2}}{4 \cdot 1}$

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

$e = 0 - \frac{4}{4 \cdot 1}$

Multiply $4$ by $1$ .

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

Divide $4$ by $4$ .

$e = 0 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Subtract $1$ from $0$ .

$e = - 1$

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

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

Substitute ${(y + 1)}^{2} - 1$ for $y^{2} + 2 y$ in the equation $4 x^{2} + y^{2} - 16 x + 2 y = - 13$ .

$4 {(x - 2)}^{2} + {(y + 1)}^{2} - 1 = - 13 + 16$

Move $- 1$ to the right side of the equation by adding $1$ to both sides.

$4 {(x - 2)}^{2} {(y + 1)}^{2} = - 13 + 16 + 1$

Add $- 13$ and $16$ .

$4 {(x - 2)}^{2} + {(y + 1)}^{2} = 3 + 1$

Add $3$ and $1$ .

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

Divide each term by $4$ to make the right side equal to one.

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

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 $1$ .

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

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