Algebra Examples

$x^{2} + 16 y^{2} = 4$

Step 1

Divide each term by $4$ to make the right side equal to one.

$\frac{x^{2}}{4} + \frac{16 y^{2}}{4} = \frac{4}{4}$

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 $1$ .

$\frac{x^{2}}{4} + \frac{y^{2}}{\frac{1}{4}} = 1$

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.

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

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.

$a = 2$

$b = \frac{1}{2}$

$k = 0$

$h = 0$

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.

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

Step 7

Simplify.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

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

$\frac{\sqrt{4 - {(\frac{1}{2})}^{2}}}{2}$

Step 7.1.2

Apply the product rule to $\frac{1}{2}$ .

$\frac{\sqrt{4 - \frac{1^{2}}{2^{2}}}}{2}$

Step 7.1.3

One to any power is one.

$\frac{\sqrt{4 - \frac{1}{2^{2}}}}{2}$

Step 7.1.4

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

$\frac{\sqrt{4 - \frac{1}{4}}}{2}$

Step 7.1.5

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\sqrt{4 \cdot \frac{4}{4} - \frac{1}{4}}}{2}$

Step 7.1.6

Combine $4$ and $\frac{4}{4}$ .

$\frac{\sqrt{\frac{4 \cdot 4}{4} - \frac{1}{4}}}{2}$

Step 7.1.7

Combine the numerators over the common denominator.

$\frac{\sqrt{\frac{4 \cdot 4 - 1}{4}}}{2}$

Step 7.1.8

Simplify the numerator.

Tap for more steps...

Step 7.1.8.1

Multiply $4$ by $4$ .

$\frac{\sqrt{\frac{16 - 1}{4}}}{2}$

Step 7.1.8.2

Subtract $1$ from $16$ .

$\frac{\sqrt{\frac{15}{4}}}{2}$

Step 7.1.9

Rewrite $\sqrt{\frac{15}{4}}$ as $\frac{\sqrt{15}}{\sqrt{4}}$ .

$\frac{\frac{\sqrt{15}}{\sqrt{4}}}{2}$

Step 7.1.10

Simplify the denominator.

Tap for more steps...

Step 7.1.10.1

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

$\frac{\frac{\sqrt{15}}{\sqrt{2^{2}}}}{2}$

Step 7.1.10.2

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

$\frac{\frac{\sqrt{15}}{2}}{2}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{15}}{2} \cdot \frac{1}{2}$

Step 7.3

Multiply $\frac{\sqrt{15}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 7.3.1

Multiply $\frac{\sqrt{15}}{2}$ by $\frac{1}{2}$ .

$\frac{\sqrt{15}}{2 \cdot 2}$

Step 7.3.2

Multiply $2$ by $2$ .

$\frac{\sqrt{15}}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{15}}{4}$

Decimal Form:

$0.96824583 \dots$

Step 9