Pre-Algebra Examples

$- 25 x^{2} + y^{2} - 100 x - 125 = 0$

Step 1

Find the standard form of the hyperbola.

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Step 1.1

Add $125$ to both sides of the equation.

$- 25 x^{2} + y^{2} - 100 x = 125$

Step 1.2

Complete the square for $- 25 x^{2} - 100 x$ .

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Step 1.2.1

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

$a = - 25$

$b = - 100$

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

$d = \frac{- 100}{2 \cdot - 25}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 100$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 100$ .

$d = \frac{2 \cdot - 50}{2 \cdot - 25}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot - 25$ .

$d = \frac{2 \cdot - 50}{2 (- 25)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 50}{2 \cdot - 25}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 50}{- 25}$

Step 1.2.3.2.2

Cancel the common factor of $- 50$ and $- 25$ .

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Step 1.2.3.2.2.1

Factor $- 25$ out of $- 50$ .

$d = \frac{- 25 (2)}{- 25}$

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $- 25$ out of $- 25$ .

$d = \frac{- 25 \cdot 2}{- 25 \cdot 1}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{- 25 \cdot 2}{- 25 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $2$ by $1$ .

$d = 2$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = 0 - \frac{{(- 100)}^{2}}{4 \cdot - 25}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Raise $- 100$ to the power of $2$ .

$e = 0 - \frac{10000}{4 \cdot - 25}$

Step 1.2.4.2.1.2

Multiply $4$ by $- 25$ .

$e = 0 - \frac{10000}{- 100}$

Step 1.2.4.2.1.3

Divide $10000$ by $- 100$ .

$e = 0 - - 100$

Step 1.2.4.2.1.4

Multiply $- 1$ by $- 100$ .

$e = 0 + 100$

Step 1.2.4.2.2

Add $0$ and $100$ .

$e = 100$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 25 {(x + 2)}^{2} + 100$ .

$- 25 {(x + 2)}^{2} + 100$

Step 1.3

Substitute $- 25 {(x + 2)}^{2} + 100$ for $- 25 x^{2} - 100 x$ in the equation $- 25 x^{2} + y^{2} - 100 x = 125$ .

$- 25 {(x + 2)}^{2} + 100 + y^{2} = 125$

Step 1.4

Move $100$ to the right side of the equation by adding $100$ to both sides.

$- 25 {(x + 2)}^{2} + y^{2} = 125 - 100$

Step 1.5

Subtract $100$ from $125$ .

$- 25 {(x + 2)}^{2} + y^{2} = 25$

Step 1.6

Divide each term by $25$ to make the right side equal to one.

$- \frac{25 {(x + 2)}^{2}}{25} + \frac{y^{2}}{25} = \frac{25}{25}$

Step 1.7

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 $1$ .

$\frac{y^{2}}{25} - \frac{{(x + 2)}^{2}}{1} = 1$

Step 2

This is the form of a hyperbola. Use this form to determine the values used to find vertices and asymptotes of the hyperbola.

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

Step 3

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, $a$ .

$a = 5$

$b = 1$

$k = 0$

$h = - 2$

Step 4

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

$(- 2, 0)$

Step 5

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

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Step 5.1

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

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

Step 5.2

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

$\sqrt{{(5)}^{2} + {(1)}^{2}}$

Step 5.3

Simplify.

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Step 5.3.1

Raise $5$ to the power of $2$ .

$\sqrt{25 + {(1)}^{2}}$

Step 5.3.2

One to any power is one.

$\sqrt{25 + 1}$

Step 5.3.3

Add $25$ and $1$ .

$\sqrt{26}$

Step 6

Find the vertices.

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Step 6.1

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

$(h, k + a)$

Step 6.2

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

$(- 2, 5)$

Step 6.3

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

$(h, k - a)$

Step 6.4

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

$(- 2, - 5)$

Step 6.5

The vertices of a hyperbola follow the form of $(h, k \pm a)$ . Hyperbolas have two vertices.

$(- 2, 5), (- 2, - 5)$

Step 7

Find the foci.

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Step 7.1

The first focus of a hyperbola can be found by adding $c$ to $k$ .

$(h, k + c)$

Step 7.2

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

$(- 2, \sqrt{26})$

Step 7.3

The second focus of a hyperbola can be found by subtracting $c$ from $k$ .

$(h, k - c)$

Step 7.4

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

$(- 2, - \sqrt{26})$

Step 7.5

The foci of a hyperbola follow the form of $(h, k \pm \sqrt{a^{2} + b^{2}})$ . Hyperbolas have two foci.

$(- 2, \sqrt{26}), (- 2, - \sqrt{26})$

Step 8

Find the eccentricity.

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Step 8.1

Find the eccentricity by using the following formula.

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

Step 8.2

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

$\frac{\sqrt{{(5)}^{2} + {(1)}^{2}}}{5}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Raise $5$ to the power of $2$ .

$\frac{\sqrt{25 + 1^{2}}}{5}$

Step 8.3.2

One to any power is one.

$\frac{\sqrt{25 + 1}}{5}$

Step 8.3.3

Add $25$ and $1$ .

$\frac{\sqrt{26}}{5}$

Step 9

Find the focal parameter.

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Step 9.1

Find the value of the focal parameter of the hyperbola by using the following formula.

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

Step 9.2

Substitute the values of $b$ and $\sqrt{a^{2} + b^{2}}$ in the formula.

$\frac{1^{2}}{\sqrt{26}}$

Step 9.3

Simplify.

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Step 9.3.1

One to any power is one.

$\frac{1}{\sqrt{26}}$

Step 9.3.2

Multiply $\frac{1}{\sqrt{26}}$ by $\frac{\sqrt{26}}{\sqrt{26}}$ .

$\frac{1}{\sqrt{26}} \cdot \frac{\sqrt{26}}{\sqrt{26}}$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply $\frac{1}{\sqrt{26}}$ by $\frac{\sqrt{26}}{\sqrt{26}}$ .

$\frac{\sqrt{26}}{\sqrt{26} \sqrt{26}}$

Step 9.3.3.2

Raise $\sqrt{26}$ to the power of $1$ .

$\frac{\sqrt{26}}{{\sqrt{26}}^{1} \sqrt{26}}$

Step 9.3.3.3

Raise $\sqrt{26}$ to the power of $1$ .

$\frac{\sqrt{26}}{{\sqrt{26}}^{1} {\sqrt{26}}^{1}}$

Step 9.3.3.4

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

$\frac{\sqrt{26}}{{\sqrt{26}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\frac{\sqrt{26}}{{\sqrt{26}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{26}}^{2}$ as $26$ .

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Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{26}$ as $26^{\frac{1}{2}}$ .

$\frac{\sqrt{26}}{{(26^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

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

$\frac{\sqrt{26}}{26^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

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

$\frac{\sqrt{26}}{26^{\frac{2}{2}}}$

Step 9.3.3.6.4

Cancel the common factor of $2$ .

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Step 9.3.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{26}}{26^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{26}}{26^{1}}$

Step 9.3.3.6.5

Evaluate the exponent.

$\frac{\sqrt{26}}{26}$

Step 10

The asymptotes follow the form $y = \pm \frac{a (x - h)}{b} + k$ because this hyperbola opens up and down.

$y = \pm 5 \cdot (x - (- 2)) + 0$

Step 11

Simplify to find the first asymptote.

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Step 11.1

Remove parentheses.

$y = 5 \cdot (x - (- 2)) + 0$

Step 11.2

Simplify $5 \cdot (x - (- 2)) + 0$ .

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Step 11.2.1

Simplify the expression.

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Step 11.2.1.1

Add $5 \cdot (x - (- 2))$ and $0$ .

$y = 5 \cdot (x - (- 2))$

Step 11.2.1.2

Multiply $- 1$ by $- 2$ .

$y = 5 \cdot (x + 2)$

Step 11.2.2

Apply the distributive property.

$y = 5 x + 5 \cdot 2$

Step 11.2.3

Multiply $5$ by $2$ .

$y = 5 x + 10$

Step 12

Simplify to find the second asymptote.

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Step 12.1

Remove parentheses.

$y = - 5 \cdot (x - (- 2)) + 0$

Step 12.2

Simplify $- 5 \cdot (x - (- 2)) + 0$ .

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Step 12.2.1

Simplify the expression.

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Step 12.2.1.1

Add $- 5 \cdot (x - (- 2))$ and $0$ .

$y = - 5 \cdot (x - (- 2))$

Step 12.2.1.2

Multiply $- 1$ by $- 2$ .

$y = - 5 \cdot (x + 2)$

Step 12.2.2

Apply the distributive property.

$y = - 5 x - 5 \cdot 2$

Step 12.2.3

Multiply $- 5$ by $2$ .

$y = - 5 x - 10$

Step 13

This hyperbola has two asymptotes.

$y = 5 x + 10, y = - 5 x - 10$

Step 14

These values represent the important values for graphing and analyzing a hyperbola.

Center: $(- 2, 0)$

Vertices: $(- 2, 5), (- 2, - 5)$

Foci: $(- 2, \sqrt{26}), (- 2, - \sqrt{26})$

Eccentricity: $\frac{\sqrt{26}}{5}$

Focal Parameter: $\frac{\sqrt{26}}{26}$

Asymptotes: $y = 5 x + 10$ , $y = - 5 x - 10$

Step 15