Rewrite each of the following expressions as a difference of two squares by completing the square (e.g. x^2 + 4x is rewritten (x + 2)^2 - 4:

(a) x^2 - 6x
(b) 3x^2 + 20x

x^2 - 6x

= x^2 - 6x + 9 - 9
= (x-3)^2 - 9

3x^2 + 20x
= 3(x^2 + (20/3)x + 100/9 - 100/9
= 3( (x + 10/3)^2 - 100/9
= 3(x+10/3)^2 - 100/3

divide 6 by 2 and square it, 9

x^2 - 6 x + 9 -9
(x-3)^2 - 3^2

3(x^2 + 20 x/3)
(10/3)^2 = 100/9

3 [ x^2 + 20/3 x + (10/3)^2 - (10/3)^2 ]

= 3 [ (x-10/3)^2 - (10/3)^2 ]

To rewrite each expression as a difference of two squares by completing the square, follow these steps:

(a) x^2 - 6x

1. Take half of the coefficient of the x-term, which is -6, and square it. (-6/2)^2 = (-3)^2 = 9.
2. Add and subtract the value obtained in step 1 to the expression: x^2 - 6x + 9 - 9.
3. Rewrite the expression, grouping the first three terms and the last term: (x^2 - 6x + 9) - 9.
4. Factor the first three terms as a perfect square: (x - 3)^2 - 9.
5. Simplify the expression: (x - 3)^2 - 9.

Therefore, the expression x^2 - 6x can be written as (x - 3)^2 - 9.

(b) 3x^2 + 20x

1. Take half of the coefficient of the x-term, which is 20, and square it. (20/2)^2 = 10^2 = 100.
2. Add and subtract the value obtained in step 1 to the expression: 3x^2 + 20x + 100 - 100.
3. Rewrite the expression, grouping the first two terms and the last term: (3x^2 + 20x + 100) - 100.
4. Factor the first three terms as a perfect square: (sqrt(3)x + 10)^2 - 100.
5. Simplify the expression: (sqrt(3)x + 10)^2 - 100.

Therefore, the expression 3x^2 + 20x can be written as (sqrt(3)x + 10)^2 - 100.

To rewrite each of the given expressions as a difference of two squares, we need to complete the square. Here's how to do it:

(a) x^2 - 6x:
To complete the square, we need to add and subtract the square of half the coefficient of the x-term. In this case, the coefficient of the x-term is -6, so we have:

x^2 - 6x + (6/2)^2 - (6/2)^2

Simplifying this expression gives us:

x^2 - 6x + 9 - 9

Now, we can rewrite it as a difference of two squares:

(x^2 - 6x + 9) - 9

Further simplifying the expression gives:

(x - 3)^2 - 9

So, the expression x^2 - 6x can be rewritten as (x - 3)^2 - 9.

(b) 3x^2 + 20x:
To complete the square, we need to add and subtract the square of half the coefficient of the x-term. In this case, the coefficient of the x-term is 20, so we have:

3x^2 + 20x + (20/2)^2 - (20/2)^2

Simplifying this expression gives us:

3x^2 + 20x + 100 - 100

Now, we can rewrite it as a difference of two squares:

(3x^2 + 20x + 100) - 100

Further simplifying the expression gives:

(3x + 10)^2 - 100

So, the expression 3x^2 + 20x can be rewritten as (3x + 10)^2 - 100.