Precalculus Examples

Step 1
Complete the square for .
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Step 1.1
Use the form , to find the values of , , and .
Step 1.2
Consider the vertex form of a parabola.
Step 1.3
Find the value of using the formula .
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Step 1.3.1
Substitute the values of and into the formula .
Step 1.3.2
Cancel the common factor of and .
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Step 1.3.2.1
Factor out of .
Step 1.3.2.2
Cancel the common factors.
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Step 1.3.2.2.1
Factor out of .
Step 1.3.2.2.2
Cancel the common factor.
Step 1.3.2.2.3
Rewrite the expression.
Step 1.3.2.2.4
Divide by .
Step 1.4
Find the value of using the formula .
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Step 1.4.1
Substitute the values of , and into the formula .
Step 1.4.2
Simplify the right side.
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Step 1.4.2.1
Simplify each term.
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Step 1.4.2.1.1
Cancel the common factor of and .
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Step 1.4.2.1.1.1
Factor out of .
Step 1.4.2.1.1.2
Cancel the common factors.
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Step 1.4.2.1.1.2.1
Factor out of .
Step 1.4.2.1.1.2.2
Cancel the common factor.
Step 1.4.2.1.1.2.3
Rewrite the expression.
Step 1.4.2.1.1.2.4
Divide by .
Step 1.4.2.1.2
Multiply by .
Step 1.4.2.2
Subtract from .
Step 1.5
Substitute the values of , , and into the vertex form .
Step 2
Substitute for in the equation .
Step 3
Move to the right side of the equation by adding to both sides.
Step 4
Complete the square for .
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Step 4.1
Use the form , to find the values of , , and .
Step 4.2
Consider the vertex form of a parabola.
Step 4.3
Find the value of using the formula .
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Step 4.3.1
Substitute the values of and into the formula .
Step 4.3.2
Simplify the right side.
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Step 4.3.2.1
Cancel the common factor of and .
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Step 4.3.2.1.1
Factor out of .
Step 4.3.2.1.2
Move the negative one from the denominator of .
Step 4.3.2.2
Rewrite as .
Step 4.3.2.3
Multiply by .
Step 4.4
Find the value of using the formula .
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Step 4.4.1
Substitute the values of , and into the formula .
Step 4.4.2
Simplify the right side.
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Step 4.4.2.1
Simplify each term.
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Step 4.4.2.1.1
Raise to the power of .
Step 4.4.2.1.2
Multiply by .
Step 4.4.2.1.3
Divide by .
Step 4.4.2.1.4
Multiply by .
Step 4.4.2.2
Add and .
Step 4.5
Substitute the values of , , and into the vertex form .
Step 5
Substitute for in the equation .
Step 6
Move to the right side of the equation by adding to both sides.
Step 7
Simplify .
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Step 7.1
Add and .
Step 7.2
Subtract from .
Step 8
Flip the sign on each term of the equation so the term on the right side is positive.
Step 9
Divide each term by to make the right side equal to one.
Step 10
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 .
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