Answer:
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.
