Please help! Giving 20 points!! I only need number 11

8

Step-by-step explanation:

How many more unit tiles must be added to the function

f(x)=x²-6x+1 in order to complete the square?

The easier, but usually not viable way: If we look at how many more tiles need to be added, we can see that each smaller sides is 1 unit. Thus, we need 8 more tiles to complete the square.

However, this is just an educated guess. If we didn't have a diagram or it was drawn badly, we would be wrong. Let's solve it algebraically.

Note:

completing the square is changing a quadratic equation (the format is

ax² + bx + c = 0)into a(x - d)² + e = 0. (an example of this is (a + b)² or (a - b)²)

This may be confusing, but look at an example:

x² + 4x + 4 = (x + 2)²

Also note that, (a - b)² = a² - 2ab + b²

If we substitute a = x, then we get:

x² - 2bx + b².

Now, all we need to do is find b.

If we know that 2bx = 6x, then we can divide both sides by 2x and get an answer of

2bx = 6x

b = 3.

Thus, b is 3.

Remember, to complete the square, we need to find b², which is

3² = 9.

Then, because they already give us a "1", we just need to add 9-1 = 8 more units.

Just in case, let's check the answer.

If we add 8 to the problem, we get:

x² - 6x + 1 + 8 =

x² - 6x + 9

As you can see, this follows the a² - 2ab + b² format, so we can factor it.

x² - 6x + 9 = (x - 3)²

This is a completed square. Thus, we need to add 8 more unit tiles.