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Teaching Completing the Square Method

Recently one of my students needed help on a "completing the square method" homework set. After a quick look at the problems I was 100% sure I had never seen this method before and was quite embarrassed that I didn't know how to help her.

I took a picture of her homework and went back home and researched the method on Google and came up with the following result:

#1:

1.) x² - 8x + 3 = 0

a. First move over the constant (in this case the 3):

i. x² - 8x = - 3

b. Then divide the whole problem by the coefficient in front of the x² term. In this case the coefficient is one so it stays how it is.

i. x² - 8x = - 3

c. Then divide the coefficient in front of the x term by 2.

i. Here the coefficient is -8 so we divide that by 2 to get -4.

ii. Next we square that number and add it to both sides (-4)²=16.

1. x² - 8x + 16 = -3 + 16

a. x² - 8x + 16 = 13 (after combining like terms)

d. Then you find the squares equation. You take the term you got at the end of step c.i which was -4 and get:

i. (x – 4) ² = 13

(This works because (x – 4) ² = (x - 4)(x-4) = x² - 8x + 16 after doing FOIL and combining like terms)

e. Take the square root of both sides.

i. v(x – 4) ² = +/- v13

ii. x – 4 = +/- v13

iii. x = 4 +/- v13

iv. x = 4 + v13, x = 4 - v13

#2) -x² + 6x – 5 = 0

-x² + 6x = 5

Now we divide both sides by the coefficient of the x² term, which here is -1 and get:

x² - 6x = -5

Divide the -6 by 2 and get -3. Then square this term and get 9. Add to both sides.

x² - 6x + 9 = -5 + 9

x² - 6x + 9 = 4

Then take the number we got by dividing the x coefficient by 2, which was -3 and get:

(x – 3)² = 4

v(x-3)² = +/- v4

x -3 = +/- 2

x = 3 +/- 2

x = 1, x = 5

#3 -3x² – 5x = -4

Divide each side by the x² coefficient (which is -3 here) and get:

X² + (5/3)x = (4/3)

Then divide the x coefficient by 2 and get:

(5/3)/2 = 5/6 then square this: (5/6)² = 25/36 and add to both sides:

X² + (5/3)x + 25/36 = (4/3) + 25/36

X² + (5/3)x + 25/36 = 73/36

Take the number we got when we divided the x coefficient by 2 (which here was 5/6) and get:

(x + 5/6)²= 73/36

v(x + 5/6)² = +/ - v73/36

X + 5/6 =+/ - v73/36

X = -5/6 +/- v73/36

X = -5/6 + v73/36 , x = -5/6 - v73/36

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