Precalculus Examples

Find the Center and Radius x^2+y^2+2x-4y+1=0
Step 1
Subtract from both sides of the equation.
Step 2
Complete the square for .
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Step 2.1
Use the form , to find the values of , , and .
Step 2.2
Consider the vertex form of a parabola.
Step 2.3
Find the value of using the formula .
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Step 2.3.1
Substitute the values of and into the formula .
Step 2.3.2
Cancel the common factor of .
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Step 2.3.2.1
Cancel the common factor.
Step 2.3.2.2
Rewrite the expression.
Step 2.4
Find the value of using the formula .
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Step 2.4.1
Substitute the values of , and into the formula .
Step 2.4.2
Simplify the right side.
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Step 2.4.2.1
Simplify each term.
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Step 2.4.2.1.1
Raise to the power of .
Step 2.4.2.1.2
Multiply by .
Step 2.4.2.1.3
Cancel the common factor of .
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Step 2.4.2.1.3.1
Cancel the common factor.
Step 2.4.2.1.3.2
Rewrite the expression.
Step 2.4.2.1.4
Multiply by .
Step 2.4.2.2
Subtract from .
Step 2.5
Substitute the values of , , and into the vertex form .
Step 3
Substitute for in the equation .
Step 4
Move to the right side of the equation by adding to both sides.
Step 5
Complete the square for .
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Step 5.1
Use the form , to find the values of , , and .
Step 5.2
Consider the vertex form of a parabola.
Step 5.3
Find the value of using the formula .
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Step 5.3.1
Substitute the values of and into the formula .
Step 5.3.2
Cancel the common factor of and .
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Step 5.3.2.1
Factor out of .
Step 5.3.2.2
Cancel the common factors.
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Step 5.3.2.2.1
Factor out of .
Step 5.3.2.2.2
Cancel the common factor.
Step 5.3.2.2.3
Rewrite the expression.
Step 5.3.2.2.4
Divide by .
Step 5.4
Find the value of using the formula .
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Step 5.4.1
Substitute the values of , and into the formula .
Step 5.4.2
Simplify the right side.
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Step 5.4.2.1
Simplify each term.
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Step 5.4.2.1.1
Cancel the common factor of and .
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Step 5.4.2.1.1.1
Rewrite as .
Step 5.4.2.1.1.2
Apply the product rule to .
Step 5.4.2.1.1.3
Raise to the power of .
Step 5.4.2.1.1.4
Multiply by .
Step 5.4.2.1.1.5
Factor out of .
Step 5.4.2.1.1.6
Cancel the common factors.
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Step 5.4.2.1.1.6.1
Factor out of .
Step 5.4.2.1.1.6.2
Cancel the common factor.
Step 5.4.2.1.1.6.3
Rewrite the expression.
Step 5.4.2.1.1.6.4
Divide by .
Step 5.4.2.1.2
Multiply by .
Step 5.4.2.2
Subtract from .
Step 5.5
Substitute the values of , , and into the vertex form .
Step 6
Substitute for in the equation .
Step 7
Move to the right side of the equation by adding to both sides.
Step 8
Simplify .
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Step 8.1
Add and .
Step 8.2
Add and .
Step 9
This is the form of a circle. Use this form to determine the center and radius of the circle.
Step 10
Match the values in this circle to those of the standard form. The variable represents the radius of the circle, represents the x-offset from the origin, and represents the y-offset from origin.
Step 11
The center of the circle is found at .
Center:
Step 12
These values represent the important values for graphing and analyzing a circle.
Center:
Radius:
Step 13