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Module 5 Topic D
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Learning Target I will be able to: Write the equations of circles and graph circles. I will be able to: Use the equation and graph of a circle to solve problems.
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Learning Target I will be able to: Complete the square in order to write the equation of a circle in center-radius form. I will be able to: recognize when a quadratic in x and y is the equation of a circle.
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Example 1 The following is the equation of a circle with radius 5 and center 1, 2 . Do you see why? π₯ 2 β2π₯+1+ π¦ 2 β4π¦+4=25 We know that the equation π₯ 2 β2π₯+1+ π¦ 2 β4π¦+4=25 is a circle with radius 5 and center (1, 2) because when we multiply out the equation π₯β π¦β2 2 = 5 2 , we get π₯ 2 β2π₯+1+ π¦ 2 β4π¦+4=25
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M5TD - Complete the Square
Rewrite the quadratic so only the first two terms are on the left of the equal sign. Make sure the leading coefficient is 1 or else divide everything by the leading coefficient to make it that way Add Β½ of the second coefficient all squared to both sides. Rewrite the left hand side as a square Square root both sides Solve x2 + 6x = 16 It already is x2 + 6x = x2 + 6x + 32 = (x + 3)2 = 25 x + 3 = ο± 5 x = -3 ο± 5 x = 2, -8
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Supplemental Information
M5TD Supplemental Information
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M5TD - Complete the Square
The Perfect Square (x + 3)2 (x + k)2 = x2 + 2kx + k2 =(x + 3)(x + 3) = x2 + 2(3x) + 32 = x2 + 6x + 9
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M5TD - Complete the Square
What should c be to make a perfect square x2 + 8x + c you have to add (8/2)2 = 42 = 16 to get a perfect square or (x + 4)2
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M5TD - Complete the Square
Rewrite the quadratic so only the first two terms are on the left of the equal sign. Make sure the leading coefficient is 1 or else divide everything by the leading coefficient to make it that way Add Β½ of the second coefficient all squared to both sides. Rewrite the left hand side as a square Square root both sides Solve x2 + 6x = 16 It already is x2 + 6x = x2 + 6x + 32 = (x + 3)2 = 25 x + 3 = ο± 5 x = -3 ο± 5 x = 2, -8
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