Given a+b=8 and a*b=12, what is the sum of squares of a and b? What is the difference a^2-b^2?

The sum of squares is:

a^2 +
b^2

If we'll complete the square, we'll
get:

a^2 + b^2 + 2ab - 2ab = (a+b)^2 - 2ab
(1)

We'll substitute in (1) the values of the sum a+b and
the value of the product a*b.

a^2 + b^2 = 8^2 -
2*12

a^2 + b^2 = 64 -
24

a^2 + b^2 =
40

Now, we'll calculate the
difference of squares:

a^2 -
b^2

We'll write the difference of squares as a
product:

a^2 - b^2 =
(a-b)(a+b)

To substitute the factor a-b by it's value,
we'll have to calkculate a and b.

We'll have the system of
equations:

a + b = 8

a = 8 - b
(2)

a*b = 12 (3)

We'll
substitute a = 8 - b in (3):

(8 - b)*b =
12

8b - b^2 = 12

We'll
subtract 12 both sides:

- b^2 + 8b - 12 =
0

We'll multiply by -1:

b^2 -
8b + 12 = 0

We'll apply the quadratic
formula:

b1 =
[8+sqrt(64-48)]/2

b1 =
(8+4)/2

b1 = 6

b2 =
2

a1 = 8 - b1

a1 = 8 -
6

a1 = 2

a2 =
6

The system is symmetric.

a -
b = 2-6 = -4

a-b = 6-2 = 4

a^2
- b^2 = (a-b)(a+b) = (-4)(8) =
-32

or

a^2 - b^2 =
32