Completing the Square
University of Minnesota
Completing the Square
Preliminaries and Objectives
Preliminaries
• Expanding binomials like (x + a)2
• General form of a circle, ellipse, parabola, hyperbola
Objectives
• Complete the square
University of Minnesota
Completing the Square
Standard Form of Conic Sections
Standard form of a circle:
(x − h)2 + (y − k )2 = r 2
Standard form of a vertical parabola:
y − k = ±a(x − h)2
Standard form of an horiz. parabola:
x − h = ±a(y − k )2
Standard form of an ellipse:
(x − h)2 (y − k )2
+
=1
a2
b2
Standard form of a horiz. hyperbola:
(x − h)2 (y − k )2
−
=1
a2
b2
Standard form of a vertical hyperbola:
(y − k )2 (x − h)2
−
=1
b2
a2
University of Minnesota
Completing the Square
Example 1
Write the circle below in standard form
y 2 + 4x = 6y − x 2 + 23
Goal : (x − h)2 + (y − k )2 = r 2
University of Minnesota
Completing the Square
Example 1
Step 1: Rearrange the terms
y 2 + 4x = 6y − x 2 + 23
x 2 + 4x +
+ y 2 − 6y +
= 23 +
Goal : (x − h)2 + (y − k )2 = r 2
University of Minnesota
Completing the Square
Example 1
Step 2: Factor out coefficients on x 2 , y 2
y 2 + 4x = 6y − x 2 + 23
x 2 + 4x +
+ y 2 − 6y +
= 23 +
Goal : (x − h)2 + (y − k )2 = r 2
University of Minnesota
Completing the Square
Example 1
Step 3: Determine perfect square
y 2 + 4x = 6y − x 2 + 23
x 2 + 4x +
+ y 2 − 6y +
= 23 +
(x + 2)(x + 2) + (y − 3)(y − 3) = 23 +
Goal : (x − h)2 + (y − k )2 = r 2
University of Minnesota
Completing the Square
Example 1
Step 4: Supply the missing constant
y 2 + 4x = 6y − x 2 + 23
x 2 + 4x + 4 + y 2 − 6y + 9 = 23 +
(x + 2)(x + 2) + (y − 3)(y − 3) = 23 +
Goal : (x − h)2 + (y − k )2 = r 2
University of Minnesota
Completing the Square
Example 1
Step 4: Supply the missing constant
y 2 + 4x = 6y − x 2 + 23
x 2 + 4x + 4 + y 2 − 6y + 9 = 23 + 4 + 9
(x + 2)(x + 2) + (y − 3)(y − 3) = 23 + 13 = 36
Goal : (x − h)2 + (y − k )2 = r 2
University of Minnesota
Completing the Square
Example 1
Step 5: Write in standard form
y 2 + 4x = 6y − x 2 + 23
x 2 + 4x + 4 + y 2 − 6y + 9 = 23 + 4 + 9
(x + 2)(x + 2) + (y − 3)(y − 3) = 23 + 13 = 36
(x + 2)2 + (y − 3)2 = 62
Goal : (x − h)2 + (y − k )2 = r 2
University of Minnesota
Completing the Square
Example 1
Step 5: Write in standard form
y 2 + 4x = 6y − x 2 + 23
x 2 + 4x + 4 + y 2 − 6y + 9 = 23 + 4 + 9
(x + 2)(x + 2) + (y − 3)(y − 3) = 23 + 13 = 36
(x + 2)2 + (y − 3)2 = 62
Goal : (x − h)2 + (y − k )2 = r 2
This is a circle with radius = 6, with center at (−2, 3)
University of Minnesota
Completing the Square
Example 2
Write the parabola below in standard form
y + x 2 = 8x − 19
Goal : y − k = A(x − h)2
University of Minnesota
Completing the Square
Example 2
Step 1: Rearrange the terms
y + x 2 = 8x − 19
y + 19 +
= −x 2 + 8x +
Goal : y − k = A(x − h)2
University of Minnesota
Completing the Square
Example 2
Step 2: Factor out coefficient on x 2
y + x 2 = 8x − 19
y + 19 +
= −x 2 + 8x +
y + 19 +
= −(x 2 − 8x +
Goal : y − k = A(x − h)2
University of Minnesota
Completing the Square
)
Example 2
Step 3: Determine perfect square
y + x 2 = 8x − 19
y + 19 +
= −x 2 + 8x +
y + 19 +
= −(x 2 − 8x +
y + 19 +
= −(x − 4)(x − 4)
Goal : y − k = A(x − h)2
University of Minnesota
Completing the Square
)
Example 2
Step 4: Supply the missing constant
y + x 2 = 8x − 19
y + 19 +
= −x 2 + 8x +
y + 19 +
= −(x 2 − 8x + 16)
y + 19 +
= −(x − 4)(x − 4)
Goal : y − k = A(x − h)2
University of Minnesota
Completing the Square
Example 2
Step 4: Supply the missing constant
y + x 2 = 8x − 19
y + 19 +
= −x 2 + 8x + − 16
y + 19 +
= −(x 2 − 8x + 16)
y + 19 +
= −(x − 4)(x − 4)
Goal : y − k = A(x − h)2
University of Minnesota
Completing the Square
Example 2
Step 4: Supply the missing constant
y + x 2 = 8x − 19
y + 19 + − 16 = −x 2 + 8x + − 16
y + 19 + − 16 = −(x 2 − 8x + 16)
y + 19 + − 16 = −(x − 4)(x − 4)
Goal : y − k = A(x − h)2
University of Minnesota
Completing the Square
Example 2
Step 5: Write in standard form
y + x 2 = 8x − 19
y + 19 + − 16 = −x 2 + 8x + − 16
y + 19 + − 16 = −(x 2 − 8x + 16)
y + 19 + − 16 = −(x − 4)(x − 4)
y + 3 = −(x − 4)2
Goal : y − k = A(x − h)2
University of Minnesota
Completing the Square
Example 2
Step 5: Write in standard form
y + x 2 = 8x − 19
y + 19 + − 16 = −x 2 + 8x + − 16
y + 19 + − 16 = −(x 2 − 8x + 16)
y + 19 + − 16 = −(x − 4)(x − 4)
y + 3 = −(x − 4)2
Goal : y − k = A(x − h)2
This is a parabola, pointed downward, with vertex at (4, −3)
University of Minnesota
Completing the Square
Example 3
Write the hyperbola below in standard form
4y 2 + 12x = 9x 2 + 4y + 39
Goal :
(y − k )2 (x − h)2
−
=1
b2
a2
University of Minnesota
Completing the Square
Example 3
Step 1: Rearrange the terms
4y 2 + 12x = 9x 2 + 4y + 39
4y 2 − 4y +
− 9x 2 + 12x +
Goal :
= 39 +
(y − k )2 (x − h)2
−
=1
b2
a2
University of Minnesota
Completing the Square
Example 3
Step 2: Factor out coefficient on x 2
4y 2 + 12x = 9x 2 + 4y + 39
4y 2 − 4y +
− 9x 2 + 12x +
= 39 +
4(y 2 − y + ) − 9(x 2 − 43 x + ) = 39 +
Goal :
(y − k )2 (x − h)2
−
=1
b2
a2
University of Minnesota
Completing the Square
Example 3
Step 3: Determine perfect square
4y 2 + 12x = 9x 2 + 4y + 39
4y 2 − 4y +
− 9x 2 + 12x +
= 39 +
4(y 2 − y + ) − 9(x 2 − 43 x + ) = 39 +
4(y − 12 )(y − 21 ) − 9(x − 23 )(x − 23 ) = 39 +
Goal :
(y − k )2 (x − h)2
−
=1
b2
a2
University of Minnesota
Completing the Square
Example 3
Step 4: Supply the missing constant
4y 2 + 12x = 9x 2 + 4y + 39
4y 2 − 4y +
− 9x 2 + 12x +
= 39 +
4(y 2 − y + 14 ) − 9(x 2 − 43 x + 49 ) = 39 +
4(y − 12 )(y − 21 ) − 9(x − 23 )(x − 23 ) = 39 +
Goal :
(y − k )2 (x − h)2
−
=1
b2
a2
University of Minnesota
Completing the Square
Example 3
Step 4: Supply the missing constant
4y 2 + 12x = 9x 2 + 4y + 39
4y 2 − 4y + 1 − 9x 2 + 12x + − 4 = 39 +
4(y 2 − y + 14 ) − 9(x 2 − 43 x + 49 ) = 39 +
4(y − 12 )(y − 21 ) − 9(x − 23 )(x − 23 ) = 39 +
Goal :
(y − k )2 (x − h)2
−
=1
b2
a2
University of Minnesota
Completing the Square
Example 3
Step 4: Supply the missing constant
4y 2 + 12x = 9x 2 + 4y + 39
4y 2 − 4y + 1 − 9x 2 + 12x + − 4 = 39 + 1 − 4
4(y 2 − y + 14 ) − 9(x 2 − 43 x + 49 ) = 39 + − 3
4(y − 12 )(y − 21 ) − 9(x − 23 )(x − 23 ) = 39 + − 3 = 36
Goal :
(y − k )2 (x − h)2
−
=1
b2
a2
University of Minnesota
Completing the Square
Example 3
Step 5: Write in standard form
4y 2 + 12x = 9x 2 + 4y + 39
4y 2 − 4y + 1 − 9x 2 + 12x + − 4 = 39 + 1 − 4
4(y 2 − y + 14 ) − 9(x 2 − 43 x + 49 ) = 39 + − 3
4(y − 12 )(y − 21 ) − 9(x − 23 )(x − 23 ) = 39 + − 3 = 36
(y − 12 )2 (x − 23 )2
−
=1
9
4
Goal :
(y − k )2 (x − h)2
−
=1
b2
a2
University of Minnesota
Completing the Square
Example 3
Step 5: Write in standard form
4y 2 + 12x = 9x 2 + 4y + 39
4y 2 − 4y + 1 − 9x 2 + 12x + − 4 = 39 + 1 − 4
4(y 2 − y + 14 ) − 9(x 2 − 43 x + 49 ) = 39 + − 3
4(y − 12 )(y − 21 ) − 9(x − 23 )(x − 23 ) = 39 + − 3 = 36
(y − 12 )2 (x − 23 )2
−
=1
9
4
Goal :
(y − k )2 (x − h)2
−
=1
b2
a2
Begin with an hyperbola with the y -axis as the transverse axis.
The asymptotes have slope 32 . The vertices are at (0, 3) and
(0, −3). This hyperbola is then shifted to the right 23 and up 21
University of Minnesota
Completing the Square
Example 3
y
x
y2 x2
−
=1
9
4
(y − 12 )2 (x − 23 )2
−
=1
9
4
University of Minnesota
Completing the Square
Recap
Completing the Square:
• Identify the general form and rearrange terms
• Factor out coefficients on x 2 and y 2
• Determine perfect square
• Supply missing constants
• Write equation in standard form
University of Minnesota
Completing the Square
Credits
Written by:
Mike Weimerskirch
Narration:
Mike Weimerskirch
Graphic Design: Mike Weimerskirch
University of Minnesota
Completing the Square
Copyright Info
c The Regents of the University of Minnesota & Mike
Weimerskirch
For a license please contact http://z.umn.edu/otc
University of Minnesota
Completing the Square