Algebra Examples

$3 x^{2} + 4 y^{2} - 6 x + 8 y - 5 = 0$

Step 1

Add $5$ to both sides of the equation.

$3 x^{2} + 4 y^{2} - 6 x + 8 y = 5$

Step 2

Complete the square for $3 x^{2} - 6 x$ .

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Step 2.1

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

$a = 3$

$b = - 6$

$c = 0$

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

$d = \frac{- 6}{2 \cdot 3}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $- 6$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $- 6$ .

$d = \frac{2 \cdot - 3}{2 \cdot 3}$

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 $2 \cdot 3$ .

$d = \frac{2 \cdot - 3}{2 (3)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 3}{2 \cdot 3}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 3}{3}$

Step 2.3.2.2

Cancel the common factor of $- 3$ and $3$ .

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Step 2.3.2.2.1

Factor $3$ out of $- 3$ .

$d = \frac{3 \cdot - 1}{3}$

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $3$ out of $3$ .

$d = \frac{3 \cdot - 1}{3 (1)}$

Step 2.3.2.2.2.2

Cancel the common factor.

$d = \frac{3 \cdot - 1}{3 \cdot 1}$

Step 2.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

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

$e = 0 - \frac{{(- 6)}^{2}}{4 \cdot 3}$

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 $- 6$ to the power of $2$ .

$e = 0 - \frac{36}{4 \cdot 3}$

Step 2.4.2.1.2

Multiply $4$ by $3$ .

$e = 0 - \frac{36}{12}$

Step 2.4.2.1.3

Divide $36$ by $12$ .

$e = 0 - 1 \cdot 3$

Step 2.4.2.1.4

Multiply $- 1$ by $3$ .

$e = 0 - 3$

Step 2.4.2.2

Subtract $3$ from $0$ .

$e = - 3$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x - 1)}^{2} - 3$ .

$3 {(x - 1)}^{2} - 3$

Step 3

Substitute $3 {(x - 1)}^{2} - 3$ for $3 x^{2} - 6 x$ in the equation $3 x^{2} + 4 y^{2} - 6 x + 8 y = 5$ .

$3 {(x - 1)}^{2} - 3 + 4 y^{2} + 8 y = 5$

Step 4

Move $- 3$ to the right side of the equation by adding $3$ to both sides.

$3 {(x - 1)}^{2} + 4 y^{2} + 8 y = 5 + 3$

Step 5

Complete the square for $4 y^{2} + 8 y$ .

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Step 5.1

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

$a = 4$

$b = 8$

$c = 0$

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

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

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $8$ and $2$ .

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Step 5.3.2.1.1

Factor $2$ out of $8$ .

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

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 $2 \cdot 4$ .

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

Step 5.3.2.1.2.2

Cancel the common factor.

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

Step 5.3.2.1.2.3

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of $4$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Rewrite the expression.

$d = 1$

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

$e = 0 - \frac{8^{2}}{4 \cdot 4}$

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 $8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot 4}$

Step 5.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{64}{16}$

Step 5.4.2.1.3

Divide $64$ by $16$ .

$e = 0 - 1 \cdot 4$

Step 5.4.2.1.4

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 5.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(y + 1)}^{2} - 4$ .

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

Step 6

Substitute $4 {(y + 1)}^{2} - 4$ for $4 y^{2} + 8 y$ in the equation $3 x^{2} + 4 y^{2} - 6 x + 8 y = 5$ .

$3 {(x - 1)}^{2} + 4 {(y + 1)}^{2} - 4 = 5 + 3$

Step 7

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

$3 {(x - 1)}^{2} + 4 {(y + 1)}^{2} = 5 + 3 + 4$

Step 8

Simplify $5 + 3 + 4$ .

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Step 8.1

Add $5$ and $3$ .

$3 {(x - 1)}^{2} + 4 {(y + 1)}^{2} = 8 + 4$

Step 8.2

Add $8$ and $4$ .

$3 {(x - 1)}^{2} + 4 {(y + 1)}^{2} = 12$

Step 9

Divide each term by $12$ to make the right side equal to one.

$\frac{3 {(x - 1)}^{2}}{12} + \frac{4 {(y + 1)}^{2}}{12} = \frac{12}{12}$

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 $1$ .

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

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