How to Complete the Square?

c^2 - 45c + 324 = 0

Add -324 to each side of the equation

c^2 - 45c + 324 - 324 = 0 - 324

c^2 - 45c + 0 = -324

c^2 - 45c = -324

The c term is -45c. Take half its coefficient (-22.5). Square it (506.25) and add it to both sides.

Add +506.25 to each side of the equation.

c^2 - 45c + 506.25 = -324 + 506.25

c^2 -45c + 506.25 = 182.25

Factor a perfect square on the left side:

(c - 22.5) (c - 22.5) = 182.25

Calculate the square root of the right side (182.25): 13.5

Break this problem into two subproblems by setting
(c - 22.5) equal to 13.5 and -13.5

Subproblem 1

c - 22.5 = 13.5

Add +22.5 to each side of the equation.

c - 22.5 + 22.5 = 13.5 + 22.5

c + 0 = 36

c = 36

Subproblem 2

c - 22.5 = -13.5

Add +22.5 to each side of the equation.

c - 22.5 + 22.5 = -13.5 + 22.5

c + 0 = 9

c = 9

Solution:

c = 36, 9

Please Note: I have tried to give you the easiest step by step method to Complete the Square. Remember to go through all the steps mentioned above. If you leave/omit one step you'll get the wrong answer. I hope this will help you and others learn how to Complete the Square. Happy New Year !

Question , unfortunately , does not make sense.

c^2-45c+324=0

(c-9)(c-36)=0 c=9, 36

c*c=c^2
-9*c=-9c
-36*c=-36c
-36*9=224

find all factors of 324 and then what factors would add up to 45

9 and 36 follows the plan

c^2 - 45c + 324 = 0
c^2 - 45c/2 - 45c/2 = -324
c^2 - 45c/2 - 45c/2 + 2025/4 = -324 + 2025/4
(c^2 - 45c/2) - (45c/2 - 2025/4) = -1296/4 + 2025/4
c(c - 45/2) - 45/2(c - 45/2) = 729/4
(c - 45/2)(c - 45/2) = 729/4
(c - 45/2)^2 = 729/4
c - 45/2 = ±√(729/4)
c - 45/2 = ±27/2
c = 45/2 ±27/2

c = 45/2 + 27/2
c = 72/2
c = 36

c = 45/2 - 27/2
c = 18/2
c = 9

∴ c = 9 , 36

I assume its 45c
c^2 - 45c +324 = 0
Factors of 324 and having sum of - 45 are -9 and -36
Sum = -9 -36 = -45 = Middle term
Product = -9 * -36 = 324 = Last term
This is done coz the general form of a quadratic equation having roots a and b is
x^2 - (a+b)x + (a*b) = 0 or
x^2 - Sx + P = 0
where S =sum of roots =a+b
P=product of roots = ab
c^2 - 45c +324 = 0
c^2 - 36c -9c +324 = 0
c(c-36) - 9(c-36) = 0
(c-9)(c-36) = 0
c = 9,36

c^2 - 45c + 324 = 0

c^2 - 45c = -324

To complete the square you need to make the LHS into the form
(ax + b)^2 - r
where r is the remainder.

So if we take
(c - 22.5)^2
we can see that i we multiply this out we get
c^2 - 45c + 506.25
we don't want the 506.25 bit, so if we have
(c - 22.5)^2 - 506.25
Then this is the same as c^2 - 45c

So we now have
(c - 22.5)^2 - 506.25 = -324

Moving the 506.25 to the other side gives

(c - 22.5)^2 = 182.25

take square roots of both sides

c - 22.5 = +or- 13.5

c = 22.5 +or- 13.5

So,
c = 36
and
c = 9

c² - 45c + 324 = 0

Take the middle term (45) and half it (45/2) and square it (45/2)² = 2025/4 and add and subtract it into the equation like this:

c² - 45c + (2025/4) - (2025/4) + 324 = 0

The first three terms make a perfect square (c - 45/2)² so:

(c - 45/2)² - (2025/4) + 324 = 0

(c - 45/2)² -729/4 = 0

(c - 45/2)² = 729/4

Take sqrt on both sides, we get:

c - 45/2 = ±√(729/4)

c - 45/2 = ±(27/2)

c = ±(27/2) + (45/2)

(i) c = (27/2) + (45/2) ==> 36 ANSWER.

(ii) c = -(27/2) + (45/2) ==> 9 ANSWER.

Hope this helps.

c^2 - 45c + 324 = 0

Factors of 324 that have a sum of -45 are -9 and -36, so simply factor it.

(c - 9)(c - 36) = 0

c - 9 = 0 or c - 36 = 0
c = 9, c = 36