Algebra Examples

$35 y^{2} - 5 x^{2} = 35$

Step 1

Divide each term by $35$ to make the right side equal to one.

$\frac{35 y^{2}}{35} - \frac{5 x^{2}}{35} = \frac{35}{35}$

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 $1$ .

$y^{2} - \frac{x^{2}}{7} = 1$

Step 3

This is the standard form of a hyperbola. Use this form to determine the eccentricity.

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

Step 4

Match the values in this hyperbola to those of the standard form. The variable $h$ represents the x-offset from the origin, $k$ represents the y-offset from origin, $a$ .

$a = 1$

$b = \sqrt{7}$

$k = 0$

$h = 0$

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.

$\frac{\sqrt{{(1)}^{2} + {(\sqrt{7})}^{2}}}{1}$

Step 7

Simplify.

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Step 7.1

Simplify the expression.

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Step 7.1.1

Divide $\sqrt{{(1)}^{2} + {(\sqrt{7})}^{2}}$ by $1$ .

$\sqrt{{(1)}^{2} + {(\sqrt{7})}^{2}}$

Step 7.1.2

One to any power is one.

$\sqrt{1 + {(\sqrt{7})}^{2}}$

Step 7.2

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 7.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{1 + {(7^{\frac{1}{2}})}^{2}}$

Step 7.2.2

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

$\sqrt{1 + 7^{\frac{1}{2} \cdot 2}}$

Step 7.2.3

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

$\sqrt{1 + 7^{\frac{2}{2}}}$

Step 7.2.4

Cancel the common factor of $2$ .

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Step 7.2.4.1

Cancel the common factor.

$\sqrt{1 + 7^{\frac{2}{2}}}$

Step 7.2.4.2

Rewrite the expression.

$\sqrt{1 + 7^{1}}$

Step 7.2.5

Evaluate the exponent.

$\sqrt{1 + 7}$

Step 7.3

Add $1$ and $7$ .

$\sqrt{8}$

Step 7.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.4.1

Factor $4$ out of $8$ .

$\sqrt{4 (2)}$

Step 7.4.2

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

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

Step 7.5

Pull terms out from under the radical.

$2 \sqrt{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2}$

Decimal Form:

$2.82842712 \dots$

Step 9