Graph y^2+2x^2+6x=10

$y^{2} + 2 x^{2} + 6 x = 10$

Find the standard form of the ellipse.

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

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

$a = 2$

$b = 6$

$c = 0$

Consider the vertex form of a parabola.

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

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

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

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

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

Factor $2$ out of $6$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{36}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 0 - \frac{36}{8}$

Cancel the common factor of $36$ and $8$ .

Factor $4$ out of $36$ .

$e = 0 - \frac{4 (9)}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

$e = 0 - \frac{9}{2}$

Subtract $\frac{9}{2}$ from $0$ .

$e = - \frac{9}{2}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + \frac{3}{2})}^{2} - \frac{9}{2}$ .

$2 {(x + \frac{3}{2})}^{2} - \frac{9}{2}$

Substitute $2 {(x + \frac{3}{2})}^{2} - \frac{9}{2}$ for $2 x^{2} + 6 x$ in the equation $y^{2} + 2 x^{2} + 6 x = 10$ .

$y^{2} + 2 {(x + \frac{3}{2})}^{2} - \frac{9}{2} = 10$

Move $- \frac{9}{2}$ to the right side of the equation by adding $\frac{9}{2}$ to both sides.

$y^{2} + 2 {(x + \frac{3}{2})}^{2} = 10 + \frac{9}{2}$

Simplify $10 + \frac{9}{2}$ .

To write $10$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y^{2} + 2 {(x + \frac{3}{2})}^{2} = 10 \cdot \frac{2}{2} + \frac{9}{2}$

Combine $10$ and $\frac{2}{2}$ .

$y^{2} + 2 {(x + \frac{3}{2})}^{2} = \frac{10 \cdot 2}{2} + \frac{9}{2}$

Combine the numerators over the common denominator.

$y^{2} + 2 {(x + \frac{3}{2})}^{2} = \frac{10 \cdot 2 + 9}{2}$

Simplify the numerator.

Multiply $10$ by $2$ .

$y^{2} + 2 {(x + \frac{3}{2})}^{2} = \frac{20 + 9}{2}$

Add $20$ and $9$ .

$y^{2} + 2 {(x + \frac{3}{2})}^{2} = \frac{29}{2}$

Divide each term by $\frac{29}{2}$ to make the right side equal to one.

$\frac{y^{2}}{\frac{29}{2}} + \frac{2 {(x + \frac{3}{2})}^{2}}{\frac{29}{2}} = \frac{\frac{29}{2}}{\frac{29}{2}}$

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 $1$ .

$\frac{y^{2}}{\frac{29}{2}} + \frac{{(x + \frac{3}{2})}^{2}}{\frac{29}{4}} = 1$

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

$\frac{{(x - h)}^{2}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1$

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = \frac{\sqrt{58}}{2}$

$b = \frac{\sqrt{29}}{2}$

$k = 0$

$h = - \frac{3}{2}$

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(- \frac{3}{2}, 0)$

Find $c$ , the distance from the center to a focus.

Find the distance from the center to a focus of the ellipse by using the following formula.

$\sqrt{a^{2} - b^{2}}$

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(\frac{\sqrt{58}}{2})}^{2} - {(\frac{\sqrt{29}}{2})}^{2}}$

Simplify.

Apply the product rule to $\frac{\sqrt{58}}{2}$ .

$\sqrt{\frac{{\sqrt{58}}^{2}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}}$

Rewrite ${\sqrt{58}}^{2}$ as $58$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{58}$ as $58^{\frac{1}{2}}$ .

$\sqrt{\frac{{(58^{\frac{1}{2}})}^{2}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{58^{\frac{1}{2} \cdot 2}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{58^{\frac{2}{2}}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{58^{\frac{2}{2}}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}}$

Rewrite the expression.

$\sqrt{\frac{58^{1}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}}$

Evaluate the exponent.

$\sqrt{\frac{58}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}}$

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

$\sqrt{\frac{58}{4} - {(\frac{\sqrt{29}}{2})}^{2}}$

Cancel the common factor of $58$ and $4$ .

Factor $2$ out of $58$ .

$\sqrt{\frac{2 (29)}{4} - {(\frac{\sqrt{29}}{2})}^{2}}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\sqrt{\frac{2 \cdot 29}{2 \cdot 2} - {(\frac{\sqrt{29}}{2})}^{2}}$

Cancel the common factor.

$\sqrt{\frac{2 \cdot 29}{2 \cdot 2} - {(\frac{\sqrt{29}}{2})}^{2}}$

Rewrite the expression.

$\sqrt{\frac{29}{2} - {(\frac{\sqrt{29}}{2})}^{2}}$

Apply the product rule to $\frac{\sqrt{29}}{2}$ .

$\sqrt{\frac{29}{2} - \frac{{\sqrt{29}}^{2}}{2^{2}}}$

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$\sqrt{\frac{29}{2} - \frac{{(29^{\frac{1}{2}})}^{2}}{2^{2}}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{29}{2} - \frac{29^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{29}{2} - \frac{29^{\frac{2}{2}}}{2^{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{29}{2} - \frac{29^{\frac{2}{2}}}{2^{2}}}$

Rewrite the expression.

$\sqrt{\frac{29}{2} - \frac{29^{1}}{2^{2}}}$

Evaluate the exponent.

$\sqrt{\frac{29}{2} - \frac{29}{2^{2}}}$

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

$\sqrt{\frac{29}{2} - \frac{29}{4}}$

To write $\frac{29}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{29}{2} \cdot \frac{2}{2} - \frac{29}{4}}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{29}{2}$ by $\frac{2}{2}$ .

$\sqrt{\frac{29 \cdot 2}{2 \cdot 2} - \frac{29}{4}}$

Multiply $2$ by $2$ .

$\sqrt{\frac{29 \cdot 2}{4} - \frac{29}{4}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{29 \cdot 2 - 29}{4}}$

Simplify the numerator.

Multiply $29$ by $2$ .

$\sqrt{\frac{58 - 29}{4}}$

Subtract $29$ from $58$ .

$\sqrt{\frac{29}{4}}$

Rewrite $\sqrt{\frac{29}{4}}$ as $\frac{\sqrt{29}}{\sqrt{4}}$ .

$\frac{\sqrt{29}}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{29}}{\sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{29}}{2}$

Find the vertices.

The first vertex of an ellipse can be found by adding $a$ to $k$ .

$(h, k + a)$

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(- \frac{3}{2}, 0 + \frac{\sqrt{58}}{2})$

Simplify.

$(- \frac{3}{2}, \frac{\sqrt{58}}{2})$

The second vertex of an ellipse can be found by subtracting $a$ from $k$ .

$(h, k - a)$

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(- \frac{3}{2}, 0 - (\frac{\sqrt{58}}{2}))$

Simplify.

$(- \frac{3}{2}, - \frac{\sqrt{58}}{2})$

Ellipses have two vertices.

${Vertex}_{1}$ : $(- \frac{3}{2}, \frac{\sqrt{58}}{2})$

${Vertex}_{2}$ : $(- \frac{3}{2}, - \frac{\sqrt{58}}{2})$

${Vertex}_{1}$ : $(- \frac{3}{2}, \frac{\sqrt{58}}{2})$

${Vertex}_{2}$ : $(- \frac{3}{2}, - \frac{\sqrt{58}}{2})$

Find the foci.

The first focus of an ellipse can be found by adding $c$ to $k$ .

$(h, k + c)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(- \frac{3}{2}, 0 + \frac{\sqrt{29}}{2})$

Simplify.

$(- \frac{3}{2}, \frac{\sqrt{29}}{2})$

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(- \frac{3}{2}, 0 - (\frac{\sqrt{29}}{2}))$

Simplify.

$(- \frac{3}{2}, - \frac{\sqrt{29}}{2})$

Ellipses have two foci.

${Focus}_{1}$ : $(- \frac{3}{2}, \frac{\sqrt{29}}{2})$

${Focus}_{2}$ : $(- \frac{3}{2}, - \frac{\sqrt{29}}{2})$

${Focus}_{1}$ : $(- \frac{3}{2}, \frac{\sqrt{29}}{2})$

${Focus}_{2}$ : $(- \frac{3}{2}, - \frac{\sqrt{29}}{2})$

Find the eccentricity.

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} - b^{2}}}{a}$

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(\frac{\sqrt{58}}{2})}^{2} - {(\frac{\sqrt{29}}{2})}^{2}}}{\frac{\sqrt{58}}{2}}$

Simplify.

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{{(\frac{\sqrt{58}}{2})}^{2} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Apply the product rule to $\frac{\sqrt{58}}{2}$ .

$\sqrt{\frac{{\sqrt{58}}^{2}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Rewrite ${\sqrt{58}}^{2}$ as $58$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{58}$ as $58^{\frac{1}{2}}$ .

$\sqrt{\frac{{(58^{\frac{1}{2}})}^{2}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{58^{\frac{1}{2} \cdot 2}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{58^{\frac{2}{2}}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{58^{\frac{2}{2}}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Rewrite the expression.

$\sqrt{\frac{58^{1}}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Evaluate the exponent.

$\sqrt{\frac{58}{2^{2}} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

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

$\sqrt{\frac{58}{4} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Cancel the common factor of $58$ and $4$ .

Factor $2$ out of $58$ .

$\sqrt{\frac{2 (29)}{4} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\sqrt{\frac{2 \cdot 29}{2 \cdot 2} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Cancel the common factor.

$\sqrt{\frac{2 \cdot 29}{2 \cdot 2} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Rewrite the expression.

$\sqrt{\frac{29}{2} - {(\frac{\sqrt{29}}{2})}^{2}} \frac{2}{\sqrt{58}}$

Apply the product rule to $\frac{\sqrt{29}}{2}$ .

$\sqrt{\frac{29}{2} - \frac{{\sqrt{29}}^{2}}{2^{2}}} \frac{2}{\sqrt{58}}$

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$\sqrt{\frac{29}{2} - \frac{{(29^{\frac{1}{2}})}^{2}}{2^{2}}} \frac{2}{\sqrt{58}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{29}{2} - \frac{29^{\frac{1}{2} \cdot 2}}{2^{2}}} \frac{2}{\sqrt{58}}$

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{29}{2} - \frac{29^{\frac{2}{2}}}{2^{2}}} \frac{2}{\sqrt{58}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{\frac{29}{2} - \frac{29^{\frac{2}{2}}}{2^{2}}} \frac{2}{\sqrt{58}}$

Rewrite the expression.

$\sqrt{\frac{29}{2} - \frac{29^{1}}{2^{2}}} \frac{2}{\sqrt{58}}$

Evaluate the exponent.

$\sqrt{\frac{29}{2} - \frac{29}{2^{2}}} \frac{2}{\sqrt{58}}$

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

$\sqrt{\frac{29}{2} - \frac{29}{4}} \frac{2}{\sqrt{58}}$

To write $\frac{29}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{29}{2} \cdot \frac{2}{2} - \frac{29}{4}} \frac{2}{\sqrt{58}}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{29}{2}$ by $\frac{2}{2}$ .

$\sqrt{\frac{29 \cdot 2}{2 \cdot 2} - \frac{29}{4}} \frac{2}{\sqrt{58}}$

Multiply $2$ by $2$ .

$\sqrt{\frac{29 \cdot 2}{4} - \frac{29}{4}} \frac{2}{\sqrt{58}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{29 \cdot 2 - 29}{4}} \frac{2}{\sqrt{58}}$

Simplify the numerator.

Multiply $29$ by $2$ .

$\sqrt{\frac{58 - 29}{4}} \frac{2}{\sqrt{58}}$

Subtract $29$ from $58$ .

$\sqrt{\frac{29}{4}} \frac{2}{\sqrt{58}}$

Rewrite $\sqrt{\frac{29}{4}}$ as $\frac{\sqrt{29}}{\sqrt{4}}$ .

$\frac{\sqrt{29}}{\sqrt{4}} \cdot \frac{2}{\sqrt{58}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{29}}{\sqrt{2^{2}}} \cdot \frac{2}{\sqrt{58}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{29}}{2} \cdot \frac{2}{\sqrt{58}}$

Simplify terms.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{29}}{2} \cdot \frac{2}{\sqrt{58}}$

Rewrite the expression.

$\sqrt{29} \frac{1}{\sqrt{58}}$

Combine $\sqrt{29}$ and $\frac{1}{\sqrt{58}}$ .

$\frac{\sqrt{29}}{\sqrt{58}}$

Combine $\sqrt{29}$ and $\sqrt{58}$ into a single radical.

$\sqrt{\frac{29}{58}}$

Cancel the common factor of $29$ and $58$ .

Factor $29$ out of $29$ .

$\sqrt{\frac{29 (1)}{58}}$

Cancel the common factors.

Factor $29$ out of $58$ .

$\sqrt{\frac{29 \cdot 1}{29 \cdot 2}}$

Cancel the common factor.

$\sqrt{\frac{29 \cdot 1}{29 \cdot 2}}$

Rewrite the expression.

$\sqrt{\frac{1}{2}}$

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\frac{\sqrt{1}}{\sqrt{2}}$

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{2}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{2}}{2^{1}}$

Evaluate the exponent.

$\frac{\sqrt{2}}{2}$

These values represent the important values for graphing and analyzing an ellipse.

Center: $(- \frac{3}{2}, 0)$

${Vertex}_{1}$ : $(- \frac{3}{2}, \frac{\sqrt{58}}{2})$

${Vertex}_{2}$ : $(- \frac{3}{2}, - \frac{\sqrt{58}}{2})$

${Focus}_{1}$ : $(- \frac{3}{2}, \frac{\sqrt{29}}{2})$

${Focus}_{2}$ : $(- \frac{3}{2}, - \frac{\sqrt{29}}{2})$

Eccentricity: $\frac{\sqrt{2}}{2}$

Graph y^2+2x^2+6x=10

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