Pre-Algebra Examples

$4 x^{2} - 16 y^{2} - 56 x + 160 y - 268 = 0$

Step 1

Find the standard form of the hyperbola.

Tap for more steps...

Step 1.1

Add $268$ to both sides of the equation.

$4 x^{2} - 16 y^{2} - 56 x + 160 y = 268$

Step 1.2

Complete the square for $4 x^{2} - 56 x$ .

Tap for more steps...

Step 1.2.1

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

$a = 4$

$b = - 56$

$c = 0$

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 $d = \frac{b}{2 a}$ .

Tap for more steps...

Step 1.2.3.1

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

$d = \frac{- 56}{2 \cdot 4}$

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $- 56$ and $2$ .

Tap for more steps...

Step 1.2.3.2.1.1

Factor $2$ out of $- 56$ .

$d = \frac{2 \cdot - 28}{2 \cdot 4}$

Step 1.2.3.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot - 28}{2 (4)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 28}{2 \cdot 4}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 28}{4}$

Step 1.2.3.2.2

Cancel the common factor of $- 28$ and $4$ .

Tap for more steps...

Step 1.2.3.2.2.1

Factor $4$ out of $- 28$ .

$d = \frac{4 \cdot - 7}{4}$

Step 1.2.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.2.2.1

Factor $4$ out of $4$ .

$d = \frac{4 \cdot - 7}{4 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{4 \cdot - 7}{4 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 7}{1}$

Step 1.2.3.2.2.2.4

Divide $- 7$ by $1$ .

$d = - 7$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$e = 0 - \frac{{(- 56)}^{2}}{4 \cdot 4}$

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

Simplify each term.

Tap for more steps...

Step 1.2.4.2.1.1

Raise $- 56$ to the power of $2$ .

$e = 0 - \frac{3136}{4 \cdot 4}$

Step 1.2.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{3136}{16}$

Step 1.2.4.2.1.3

Divide $3136$ by $16$ .

$e = 0 - 1 \cdot 196$

Step 1.2.4.2.1.4

Multiply $- 1$ by $196$ .

$e = 0 - 196$

Step 1.2.4.2.2

Subtract $196$ from $0$ .

$e = - 196$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - 7)}^{2} - 196$ .

$4 {(x - 7)}^{2} - 196$

Step 1.3

Substitute $4 {(x - 7)}^{2} - 196$ for $4 x^{2} - 56 x$ in the equation $4 x^{2} - 16 y^{2} - 56 x + 160 y = 268$ .

$4 {(x - 7)}^{2} - 196 - 16 y^{2} + 160 y = 268$

Step 1.4

Move $- 196$ to the right side of the equation by adding $196$ to both sides.

$4 {(x - 7)}^{2} - 16 y^{2} + 160 y = 268 + 196$

Step 1.5

Complete the square for $- 16 y^{2} + 160 y$ .

Tap for more steps...

Step 1.5.1

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

$a = - 16$

$b = 160$

$c = 0$

Step 1.5.2

Consider the vertex form of a parabola.

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

Step 1.5.3

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

Tap for more steps...

Step 1.5.3.1

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

$d = \frac{160}{2 \cdot - 16}$

Step 1.5.3.2

Simplify the right side.

Tap for more steps...

Step 1.5.3.2.1

Cancel the common factor of $160$ and $2$ .

Tap for more steps...

Step 1.5.3.2.1.1

Factor $2$ out of $160$ .

$d = \frac{2 \cdot 80}{2 \cdot - 16}$

Step 1.5.3.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.5.3.2.1.2.1

Factor $2$ out of $2 \cdot - 16$ .

$d = \frac{2 \cdot 80}{2 (- 16)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 80}{2 \cdot - 16}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{80}{- 16}$

Step 1.5.3.2.2

Cancel the common factor of $80$ and $- 16$ .

Tap for more steps...

Step 1.5.3.2.2.1

Factor $16$ out of $80$ .

$d = \frac{16 (5)}{- 16}$

Step 1.5.3.2.2.2

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$d = - 1 \cdot 5$

Step 1.5.3.2.3

Multiply $- 1$ by $5$ .

$d = - 5$

Step 1.5.4

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

Tap for more steps...

Step 1.5.4.1

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

$e = 0 - \frac{160^{2}}{4 \cdot - 16}$

Step 1.5.4.2

Simplify the right side.

Tap for more steps...

Step 1.5.4.2.1

Simplify each term.

Tap for more steps...

Step 1.5.4.2.1.1

Raise $160$ to the power of $2$ .

$e = 0 - \frac{25600}{4 \cdot - 16}$

Step 1.5.4.2.1.2

Multiply $4$ by $- 16$ .

$e = 0 - \frac{25600}{- 64}$

Step 1.5.4.2.1.3

Divide $25600$ by $- 64$ .

$e = 0 - - 400$

Step 1.5.4.2.1.4

Multiply $- 1$ by $- 400$ .

$e = 0 + 400$

Step 1.5.4.2.2

Add $0$ and $400$ .

$e = 400$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 16 {(y - 5)}^{2} + 400$ .

$- 16 {(y - 5)}^{2} + 400$

Step 1.6

Substitute $- 16 {(y - 5)}^{2} + 400$ for $- 16 y^{2} + 160 y$ in the equation $4 x^{2} - 16 y^{2} - 56 x + 160 y = 268$ .

$4 {(x - 7)}^{2} - 16 {(y - 5)}^{2} + 400 = 268 + 196$

Step 1.7

Move $400$ to the right side of the equation by adding $400$ to both sides.

$4 {(x - 7)}^{2} - 16 {(y - 5)}^{2} = 268 + 196 - 400$

Step 1.8

Simplify $268 + 196 - 400$ .

Tap for more steps...

Step 1.8.1

Add $268$ and $196$ .

$4 {(x - 7)}^{2} - 16 {(y - 5)}^{2} = 464 - 400$

Step 1.8.2

Subtract $400$ from $464$ .

$4 {(x - 7)}^{2} - 16 {(y - 5)}^{2} = 64$

Step 1.9

Divide each term by $64$ to make the right side equal to one.

$\frac{4 {(x - 7)}^{2}}{64} - \frac{16 {(y - 5)}^{2}}{64} = \frac{64}{64}$

Step 1.10

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 $1$ .

$\frac{{(x - 7)}^{2}}{16} - \frac{{(y - 5)}^{2}}{4} = 1$

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.

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

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, $a$ .

$a = 4$

$b = 2$

$k = 5$

$h = 7$

Step 4

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

$(7, 5)$

Step 5

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

Tap for more steps...

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.

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

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

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

$\sqrt{16 + {(2)}^{2}}$

Step 5.3.2

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

$\sqrt{16 + 4}$

Step 5.3.3

Add $16$ and $4$ .

$\sqrt{20}$

Step 5.3.4

Rewrite $20$ as $2^{2} \cdot 5$ .

Tap for more steps...

Step 5.3.4.1

Factor $4$ out of $20$ .

$\sqrt{4 (5)}$

Step 5.3.4.2

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

$\sqrt{2^{2} \cdot 5}$

Step 5.3.5

Pull terms out from under the radical.

$2 \sqrt{5}$

Step 6

Find the vertices.

Tap for more steps...

Step 6.1

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

$(h + a, k)$

Step 6.2

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

$(11, 5)$

Step 6.3

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

$(h - a, k)$

Step 6.4

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

$(3, 5)$

Step 6.5

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

$(11, 5), (3, 5)$

Step 7

Find the foci.

Tap for more steps...

Step 7.1

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

$(h + c, k)$

Step 7.2

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

$(7 + 2 \sqrt{5}, 5)$

Step 7.3

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

$(h - c, k)$

Step 7.4

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

$(7 - 2 \sqrt{5}, 5)$

Step 7.5

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

$(7 + 2 \sqrt{5}, 5), (7 - 2 \sqrt{5}, 5)$

Step 8

Find the eccentricity.

Tap for more steps...

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.

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

Step 8.3

Simplify.

Tap for more steps...

Step 8.3.1

Simplify the numerator.

Tap for more steps...

Step 8.3.1.1

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

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

Step 8.3.1.2

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

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

Step 8.3.1.3

Add $16$ and $4$ .

$\frac{\sqrt{20}}{4}$

Step 8.3.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

Tap for more steps...

Step 8.3.1.4.1

Factor $4$ out of $20$ .

$\frac{\sqrt{4 (5)}}{4}$

Step 8.3.1.4.2

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

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

Step 8.3.1.5

Pull terms out from under the radical.

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

Step 8.3.2

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 8.3.2.1

Factor $2$ out of $2 \sqrt{5}$ .

$\frac{2 (\sqrt{5})}{4}$

Step 8.3.2.2

Cancel the common factors.

Tap for more steps...

Step 8.3.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \sqrt{5}}{2 \cdot 2}$

Step 8.3.2.2.2

Cancel the common factor.

$\frac{2 \sqrt{5}}{2 \cdot 2}$

Step 8.3.2.2.3

Rewrite the expression.

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

Step 9

Find the focal parameter.

Tap for more steps...

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.

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

Step 9.3

Simplify.

Tap for more steps...

Step 9.3.1

Cancel the common factor of $2^{2}$ and $2$ .

Tap for more steps...

Step 9.3.1.1

Factor $2$ out of $2^{2}$ .

$\frac{2 \cdot 2}{2 \sqrt{5}}$

Step 9.3.1.2

Cancel the common factors.

Tap for more steps...

Step 9.3.1.2.1

Factor $2$ out of $2 \sqrt{5}$ .

$\frac{2 \cdot 2}{2 (\sqrt{5})}$

Step 9.3.1.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \sqrt{5}}$

Step 9.3.1.2.3

Rewrite the expression.

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

Step 9.3.2

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 9.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 9.3.3.1

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 9.3.3.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 9.3.3.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 9.3.3.4

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

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

Step 9.3.3.5

Add $1$ and $1$ .

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

Step 9.3.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Tap for more steps...

Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 9.3.3.6.2

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

$\frac{2 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

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

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

Step 9.3.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.3.3.6.4.1

Cancel the common factor.

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

Step 9.3.3.6.4.2

Rewrite the expression.

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

Step 9.3.3.6.5

Evaluate the exponent.

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

Step 10

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

$y = \pm \frac{1}{2} \cdot (x - (7)) + 5$

Step 11

Simplify to find the first asymptote.

Tap for more steps...

Step 11.1

Remove parentheses.

$y = \frac{1}{2} \cdot (x - (7)) + 5$

Step 11.2

Simplify $\frac{1}{2} \cdot (x - (7)) + 5$ .

Tap for more steps...

Step 11.2.1

Simplify each term.

Tap for more steps...

Step 11.2.1.1

Multiply $- 1$ by $7$ .

$y = \frac{1}{2} \cdot (x - 7) + 5$

Step 11.2.1.2

Apply the distributive property.

$y = \frac{1}{2} x + \frac{1}{2} \cdot - 7 + 5$

Step 11.2.1.3

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

$y = \frac{x}{2} + \frac{1}{2} \cdot - 7 + 5$

Step 11.2.1.4

Combine $\frac{1}{2}$ and $- 7$ .

$y = \frac{x}{2} + \frac{- 7}{2} + 5$

Step 11.2.1.5

Move the negative in front of the fraction.

$y = \frac{x}{2} - \frac{7}{2} + 5$

Step 11.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{x}{2} - \frac{7}{2} + 5 \cdot \frac{2}{2}$

Step 11.2.3

Combine $5$ and $\frac{2}{2}$ .

$y = \frac{x}{2} - \frac{7}{2} + \frac{5 \cdot 2}{2}$

Step 11.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{2} + \frac{- 7 + 5 \cdot 2}{2}$

Step 11.2.5

Simplify the numerator.

Tap for more steps...

Step 11.2.5.1

Multiply $5$ by $2$ .

$y = \frac{x}{2} + \frac{- 7 + 10}{2}$

Step 11.2.5.2

Add $- 7$ and $10$ .

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

Step 12

Simplify to find the second asymptote.

Tap for more steps...

Step 12.1

Remove parentheses.

$y = - \frac{1}{2} \cdot (x - (7)) + 5$

Step 12.2

Simplify $- \frac{1}{2} \cdot (x - (7)) + 5$ .

Tap for more steps...

Step 12.2.1

Simplify each term.

Tap for more steps...

Step 12.2.1.1

Multiply $- 1$ by $7$ .

$y = - \frac{1}{2} \cdot (x - 7) + 5$

Step 12.2.1.2

Apply the distributive property.

$y = - \frac{1}{2} x - \frac{1}{2} \cdot - 7 + 5$

Step 12.2.1.3

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

$y = - \frac{x}{2} - \frac{1}{2} \cdot - 7 + 5$

Step 12.2.1.4

Multiply $- \frac{1}{2} \cdot - 7$ .

Tap for more steps...

Step 12.2.1.4.1

Multiply $- 7$ by $- 1$ .

$y = - \frac{x}{2} + 7 (\frac{1}{2}) + 5$

Step 12.2.1.4.2

Combine $7$ and $\frac{1}{2}$ .

$y = - \frac{x}{2} + \frac{7}{2} + 5$

Step 12.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{x}{2} + \frac{7}{2} + 5 \cdot \frac{2}{2}$

Step 12.2.3

Combine $5$ and $\frac{2}{2}$ .

$y = - \frac{x}{2} + \frac{7}{2} + \frac{5 \cdot 2}{2}$

Step 12.2.4

Combine the numerators over the common denominator.

$y = - \frac{x}{2} + \frac{7 + 5 \cdot 2}{2}$

Step 12.2.5

Simplify the numerator.

Tap for more steps...

Step 12.2.5.1

Multiply $5$ by $2$ .

$y = - \frac{x}{2} + \frac{7 + 10}{2}$

Step 12.2.5.2

Add $7$ and $10$ .

$y = - \frac{x}{2} + \frac{17}{2}$

Step 13

This hyperbola has two asymptotes.

$y = \frac{x}{2} + \frac{3}{2}, y = - \frac{x}{2} + \frac{17}{2}$

Step 14

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

Center: $(7, 5)$

Vertices: $(11, 5), (3, 5)$

Foci: $(7 + 2 \sqrt{5}, 5), (7 - 2 \sqrt{5}, 5)$

Eccentricity: $\frac{\sqrt{5}}{2}$

Focal Parameter: $\frac{2 \sqrt{5}}{5}$

Asymptotes: $y = \frac{x}{2} + \frac{3}{2}$ , $y = - \frac{x}{2} + \frac{17}{2}$

Step 15