Completing the square

Question 1

When is completing the square used?

Answer 1

To find the max/min point on a curve or its line of symmetry

Question 2

Example 1: Write x^2+8x+6 in the form (x+a)^2+b, where a and b are constants

Answer 2

First of all, constants just means numbers. Also, (x+a)^2+b means complete the square.
x^2+8x+6
1. Half the x number (e.g. 8):
8/2=4. This is your a value
2. Write the a value in: (x+4)^2-4^2+b. Notice how we minus 4^2, we always minus the a value squared after the brackets.
3. Write the extra number on the end (the b bit is also the extra number without the x, in this case a 6):
(x+4)^2-4^2+6
4. Simplify and solve:
(x+4)^2-16+6
(x+4)^2-10
a=4
b=-10

Question 3

Example 2: x^2-6x-3=(x-a)^2-b where a and b are constants. Find the values of a and b.

Answer 3

x^2-6x-3
1) -6/2=-3
a=-3
2) (x-3)^2-(-3)^2
3) (x-3)^2-(-3)^2-3
4) (x-3)^2-9-3
(x-3)^2-12
a=-3, b=-12

Question 4

Example 3: Write in the form (x+a)^2+b. Find the values of and a b: x^2-9x+3

Answer 4

x^2-9x+3
Method 1 (calculator):
1) (x-4.5)^2-(4.5)^2
2) (x-4.5)^2-(4.5)^2+3
3) (x-4.5)^2-20.25+3
4) (x-4.5)^2-17.25
5) a=-4.5, b=-17.25
Method 2 (non-calculator):
1) (x-9/2)^2-(-9/2)^2
2) (x-9/2)^2-(-9/2)^2+3
3) (x-9/2)^2-81/4+3
4) (x-9/2)^2-81/4+3/1
5) (x-9/2)^2-81/4+12/4
6) (x-9/2)^2-69/4
7) a=-9/2, b=-69/4