Algebra Examples

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

Add $5$ to both sides of the equation.

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

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

Tap for more steps...

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

$a = 3, b = - 6, c = 0$

Consider the vertex form of a parabola.

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

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

$d = - \frac{6}{2 (3)}$

Simplify the right side.

Tap for more steps...

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

Tap for more steps...

Factor $2$ out of $6$ .

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

Cancel the common factors.

Tap for more steps...

Factor $2$ out of $2 \cdot 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Tap for more steps...

Cancel the common factor.

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

Divide $1$ by $1$ .

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$d = - 1$

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

Tap for more steps...

Simplify each term.

Tap for more steps...

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

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

Multiply $4$ by $3$ .

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

Divide $36$ by $12$ .

$e = 0 - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$e = 0 - 3$

Subtract $3$ from $0$ .

$e = - 3$

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

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

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

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

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

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

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

Tap for more steps...

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

$a = 4, b = 8, c = 0$

Consider the vertex form of a parabola.

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

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

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

Simplify the right side.

Tap for more steps...

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

Tap for more steps...

Factor $2$ out of $8$ .

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

Cancel the common factors.

Tap for more steps...

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $4$ .

Tap for more steps...

Cancel the common factor.

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

Divide $1$ by $1$ .

$d = 1$

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

Tap for more steps...

Simplify each term.

Tap for more steps...

Raise $8$ to the power of $2$ .

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

Multiply $4$ by $4$ .

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

Divide $64$ by $16$ .

$e = 0 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Subtract $4$ from $0$ .

$e = - 4$

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

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

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

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

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

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

Simplify $5 + 3 + 4$ .

Tap for more steps...

Add $5$ and $3$ .

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

Add $8$ and $4$ .

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

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

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

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 $1$ .

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

Enter YOUR Problem