Algebra Examples

$y^{2} = 1 - 3 x^{2}$

Step 1

Move all terms containing variables to the left side of the equation.

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Step 1.1

Add $3 x^{2}$ to both sides of the equation.

$y^{2} + 3 x^{2} = 1$

Step 1.2

Reorder $y^{2}$ and $3 x^{2}$ .

$3 x^{2} + y^{2} = 1$

Step 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 $1$ .

$\frac{x^{2}}{\frac{1}{3}} + y^{2} = 1$

Step 3

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.

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

Step 4

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.

$a = 1$

$b = \frac{\sqrt{3}}{3}$

$k = 0$

$h = 0$

Step 5

Find the eccentricity by using the following formula.

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

Step 6

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

$\frac{\sqrt{{(1)}^{2} - {(\frac{\sqrt{3}}{3})}^{2}}}{1}$

Step 7

Simplify.

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Step 7.1

Simplify the expression.

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Step 7.1.1

Divide $\sqrt{{(1)}^{2} - {(\frac{\sqrt{3}}{3})}^{2}}$ by $1$ .

$\sqrt{{(1)}^{2} - {(\frac{\sqrt{3}}{3})}^{2}}$

Step 7.1.2

One to any power is one.

$\sqrt{1 - {(\frac{\sqrt{3}}{3})}^{2}}$

Step 7.1.3

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

$\sqrt{1 - \frac{{\sqrt{3}}^{2}}{3^{2}}}$

Step 7.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{1 - \frac{{(3^{\frac{1}{2}})}^{2}}{3^{2}}}$

Step 7.2.2

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

$\sqrt{1 - \frac{3^{\frac{1}{2} \cdot 2}}{3^{2}}}$

Step 7.2.3

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

$\sqrt{1 - \frac{3^{\frac{2}{2}}}{3^{2}}}$

Step 7.2.4

Cancel the common factor of $2$ .

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Step 7.2.4.1

Cancel the common factor.

$\sqrt{1 - \frac{3^{\frac{2}{2}}}{3^{2}}}$

Step 7.2.4.2

Rewrite the expression.

$\sqrt{1 - \frac{3^{1}}{3^{2}}}$

Step 7.2.5

Evaluate the exponent.

$\sqrt{1 - \frac{3}{3^{2}}}$

Step 7.3

Raise $3$ to the power of $2$ .

$\sqrt{1 - \frac{3}{9}}$

Step 7.4

Cancel the common factor of $3$ and $9$ .

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Step 7.4.1

Factor $3$ out of $3$ .

$\sqrt{1 - \frac{3 (1)}{9}}$

Step 7.4.2

Cancel the common factors.

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Step 7.4.2.1

Factor $3$ out of $9$ .

$\sqrt{1 - \frac{3 \cdot 1}{3 \cdot 3}}$

Step 7.4.2.2

Cancel the common factor.

$\sqrt{1 - \frac{3 \cdot 1}{3 \cdot 3}}$

Step 7.4.2.3

Rewrite the expression.

$\sqrt{1 - \frac{1}{3}}$

Step 7.5

Simplify the expression.

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Step 7.5.1

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{3}{3} - \frac{1}{3}}$

Step 7.5.2

Combine the numerators over the common denominator.

$\sqrt{\frac{3 - 1}{3}}$

Step 7.5.3

Subtract $1$ from $3$ .

$\sqrt{\frac{2}{3}}$

Step 7.6

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

$\frac{\sqrt{2}}{\sqrt{3}}$

Step 7.7

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 7.8

Combine and simplify the denominator.

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Step 7.8.1

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.8.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 7.8.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 7.8.4

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

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 7.8.5

Add $1$ and $1$ .

$\frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 7.8.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 7.8.6.2

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

$\frac{\sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 7.8.6.3

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

$\frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.8.6.4

Cancel the common factor of $2$ .

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Step 7.8.6.4.1

Cancel the common factor.

$\frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.8.6.4.2

Rewrite the expression.

$\frac{\sqrt{2} \sqrt{3}}{3^{1}}$

Step 7.8.6.5

Evaluate the exponent.

$\frac{\sqrt{2} \sqrt{3}}{3}$

Step 7.9

Simplify the numerator.

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Step 7.9.1

Combine using the product rule for radicals.

$\frac{\sqrt{2 \cdot 3}}{3}$

Step 7.9.2

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6}}{3}$

Decimal Form:

$0.81649658 \dots$

Step 9