Algebra Examples

$9 x^{2} - 16 y^{2} = 144$

Step 1

Divide each term by $144$ to make the right side equal to one.

$\frac{9 x^{2}}{144} - \frac{16 y^{2}}{144} = \frac{144}{144}$

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 $1$ .

$\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$

Step 3

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

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

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, $a$ .

$a = 4$

$b = 3$

$k = 0$

$h = 0$

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.

$\frac{\sqrt{{(4)}^{2} + {(3)}^{2}}}{4}$

Step 7

Simplify the numerator.

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Step 7.1

Raise $4$ to the power of $2$ .

$\frac{\sqrt{16 + 3^{2}}}{4}$

Step 7.2

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

$\frac{\sqrt{16 + 9}}{4}$

Step 7.3

Add $16$ and $9$ .

$\frac{\sqrt{25}}{4}$

Step 7.4

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}}}{4}$

Step 7.5

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

$\frac{5}{4}$

Step 8