| Step 1 - Understanding the Problem | |
| Let's start by reviewing how to square a binomial. (hint use FOIL: First-Outer-Inner-Last) | |
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(a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2 |
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| Notice that the middle term of a2 + 2ab + b2 is equal to 2 times the product of the first term , a , and last term , b , of the original binomial. 2ab | |
| So to complete the square, we
want to perform the opposite operation on the expression. Take half the coefficient of the
middle term ....and square it Here is the proof: |
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Show this is true! |
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| Step 2 - Complete the Square | |
| Example 1 | |
| We want to
convert the expression x2 +
6x into
a perfect square. Take half the coefficient of the middle term ....and square it |
|
| x2 +
6x
+ (6/2)2
= x2 + 6x + 32
= x2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)2 |
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| Example 2 | |
| Convert the expression x2 + 10x into a perfect square. | |
| x2 +
10x
+ (10/2)2
= x2 + 10x + 52
= x2 + 10x + 25 = (x + 5)(x + 5) = (x + 5)2 |
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| Example 3 | |
| Convert the expression x2 - 5x into a perfect square. | |
| x2
- 5x
+ (5/2)2
= x2 - 5x + (5/2)2
= x2 - 5x + 25/4 = (x - 5/2)(x - 5/2) = (x - 5/2)2 |
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| Notice: (a - b)2 = (a - b)(a - b) = a2 - ab - ab + b2 = a2 - 2ab + b2 |
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| Example 4 | |
| Convert the expression x2 - 5x into a perfect square. | |
| x2
- 5x
+ (5/2)2
= x2 - 5x + (5/2)2
= x2 - 5x + 25/4 = (x - 5/2)(x - 5/2) = (x - 5/2)2 |
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| Example 5 - Standard form of a Circle | |
| Given: x2
+ y2 + 10x - 3y + 3 = 0 Write in standard form: (x - h)2 + (x - k)2 = r2 |
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| x2 + y2 + 10x - 3y + 3 = 0 | Group like terms |
| x2 + 10x + y2 - 3y = -3 | Reorder and subtract 3 |
| x2 + 10x + 52 + y2 - 3y + (3/2)2 = -3 + 52 + (3/2)2 | Take half the middle term and square it. Note half of 3 is 3/2 |
| (x + 5)2 + (x - 3/2)2 = -3 + 25 + 9/4 | Complete the squares |
| (x + 5)2 + (x - 3/2)2 = 107/4 | |
| Here the center is (-5,3/2) | Radius = sqrt(107)/2 |
