Pre-Algebra Examples

Graph -25x^2+y^2-100x-125=0
Step 1
Find the standard form of the hyperbola.
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Step 1.1
Add to both sides of the equation.
Step 1.2
Complete the square for .
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Step 1.2.1
Use the form , to find the values of , , and .
Step 1.2.2
Consider the vertex form of a parabola.
Step 1.2.3
Find the value of using the formula .
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Step 1.2.3.1
Substitute the values of and into the formula .
Step 1.2.3.2
Simplify the right side.
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Step 1.2.3.2.1
Cancel the common factor of and .
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Step 1.2.3.2.1.1
Factor out of .
Step 1.2.3.2.1.2
Cancel the common factors.
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Step 1.2.3.2.1.2.1
Factor out of .
Step 1.2.3.2.1.2.2
Cancel the common factor.
Step 1.2.3.2.1.2.3
Rewrite the expression.
Step 1.2.3.2.2
Cancel the common factor of and .
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Step 1.2.3.2.2.1
Factor out of .
Step 1.2.3.2.2.2
Cancel the common factors.
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Step 1.2.3.2.2.2.1
Factor out of .
Step 1.2.3.2.2.2.2
Cancel the common factor.
Step 1.2.3.2.2.2.3
Rewrite the expression.
Step 1.2.3.2.2.2.4
Divide by .
Step 1.2.4
Find the value of using the formula .
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Step 1.2.4.1
Substitute the values of , and into the formula .
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 to the power of .
Step 1.2.4.2.1.2
Multiply by .
Step 1.2.4.2.1.3
Divide by .
Step 1.2.4.2.1.4
Multiply by .
Step 1.2.4.2.2
Add and .
Step 1.2.5
Substitute the values of , , and into the vertex form .
Step 1.3
Substitute for in the equation .
Step 1.4
Move to the right side of the equation by adding to both sides.
Step 1.5
Subtract from .
Step 1.6
Divide each term by to make the right side equal to one.
Step 1.7
Simplify each term in the equation in order to set the right side equal to . The standard form of an ellipse or hyperbola requires the right side of the equation be .
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.
Step 3
Match the values in this hyperbola to those of the standard form. The variable represents the x-offset from the origin, represents the y-offset from origin, .
Step 4
The center of a hyperbola follows the form of . Substitute in the values of and .
Step 5
Find , 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.
Step 5.2
Substitute the values of and in the formula.
Step 5.3
Simplify.
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Step 5.3.1
Raise to the power of .
Step 5.3.2
One to any power is one.
Step 5.3.3
Add and .
Step 6
Find the vertices.
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Step 6.1
The first vertex of a hyperbola can be found by adding to .
Step 6.2
Substitute the known values of , , and into the formula and simplify.
Step 6.3
The second vertex of a hyperbola can be found by subtracting from .
Step 6.4
Substitute the known values of , , and into the formula and simplify.
Step 6.5
The vertices of a hyperbola follow the form of . Hyperbolas have two vertices.
Step 7
Find the foci.
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Step 7.1
The first focus of a hyperbola can be found by adding to .
Step 7.2
Substitute the known values of , , and into the formula and simplify.
Step 7.3
The second focus of a hyperbola can be found by subtracting from .
Step 7.4
Substitute the known values of , , and into the formula and simplify.
Step 7.5
The foci of a hyperbola follow the form of . Hyperbolas have two foci.
Step 8
Find the eccentricity.
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Step 8.1
Find the eccentricity by using the following formula.
Step 8.2
Substitute the values of and into the formula.
Step 8.3
Simplify the numerator.
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Step 8.3.1
Raise to the power of .
Step 8.3.2
One to any power is one.
Step 8.3.3
Add and .
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.
Step 9.2
Substitute the values of and in the formula.
Step 9.3
Simplify.
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Step 9.3.1
One to any power is one.
Step 9.3.2
Multiply by .
Step 9.3.3
Combine and simplify the denominator.
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Step 9.3.3.1
Multiply by .
Step 9.3.3.2
Raise to the power of .
Step 9.3.3.3
Raise to the power of .
Step 9.3.3.4
Use the power rule to combine exponents.
Step 9.3.3.5
Add and .
Step 9.3.3.6
Rewrite as .
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Step 9.3.3.6.1
Use to rewrite as .
Step 9.3.3.6.2
Apply the power rule and multiply exponents, .
Step 9.3.3.6.3
Combine and .
Step 9.3.3.6.4
Cancel the common factor of .
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Step 9.3.3.6.4.1
Cancel the common factor.
Step 9.3.3.6.4.2
Rewrite the expression.
Step 9.3.3.6.5
Evaluate the exponent.
Step 10
The asymptotes follow the form because this hyperbola opens up and down.
Step 11
Simplify to find the first asymptote.
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Step 11.1
Remove parentheses.
Step 11.2
Simplify .
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Step 11.2.1
Simplify the expression.
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Step 11.2.1.1
Add and .
Step 11.2.1.2
Multiply by .
Step 11.2.2
Apply the distributive property.
Step 11.2.3
Multiply by .
Step 12
Simplify to find the second asymptote.
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Step 12.1
Remove parentheses.
Step 12.2
Simplify .
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Step 12.2.1
Simplify the expression.
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Step 12.2.1.1
Add and .
Step 12.2.1.2
Multiply by .
Step 12.2.2
Apply the distributive property.
Step 12.2.3
Multiply by .
Step 13
This hyperbola has two asymptotes.
Step 14
These values represent the important values for graphing and analyzing a hyperbola.
Center:
Vertices:
Foci:
Eccentricity:
Focal Parameter:
Asymptotes: ,
Step 15