Answer:
1. (x - 3)² = 21
2. (x - 2)² = 22
3. (x - 1)² = -7
4. (x + 6)² = 27
Step-by-step explanation:
Just in case you still need the answer or want to know the process for this, it's the same couple of steps every time; once you memorize the steps, it's smooth sailing.
First, I want to note that the general form of equations like this is Ax² + Bx + C = 0.
1. x² - 6x - 12 = 0
There are a couple of different ways, but they all do the same thing. I'll show you my way. The first thing I do every time I have to complete the square is move the constant value ("C") to the right of the equal sign:
x² - 6x = 12,
so I can visualize that space where I need to put a perfect square to make sure I can get into the form (x - p)². The way you find that "p" value in the formula is by taking the middle value ("B") of the equation we wrote above, dividing it by two, then squaring it. Like this:
- 6 ÷ 2 = -3
(-3)² = 9,
and you take that answer and add it to BOTH sides of the equation:
x² - 6x + 9 = 12 + 9,
add like terms,
x² - 6x + 9 = 21,
and, finally, rearrange the left side of the equation to fit the (x - p)² form. This number is always your middle value ("B") divided by 2 (the value we found earlier when we did -6 ÷ 2 = -3).
(x - 3)² = 21.
You can always double check this by quickly expanding/FOIL-ing the left side to make sure you get the values -6 in the middle and 9 as the constant.
So if we want to summarize those steps...
1. Move the constant to the other side of the equation. (Sometimes it helps to leave an empty space on the left side so you remember you have to add something to both sides.)
2. Take the coefficient of the variable "x", or your "middle number," or your "B" divide it by 2, and square it. That gives you your new "C".
3. Add the result of #2 to both sides of the equation.
4. Simplify and factor.
With those down, try #2.
2. x² - 4x - 18 = 0
Step 1, move the constant.
x² - 4x = 18
Step 2, take "B", split it, then square it.
-4 ÷2 = -2
(-2)² = 4
Step 3, add it to both sides of the equation.
x² - 4x + 4 = 18 + 4
Step 4, simplify/add like terms and factor.
(x - 2)² = 22.
And the other two follow similarly.
3. x² - 2x + 8 = 0
x² - 2x = -8
x² - 2x + 1 = -8 + 1
(x - 1)² = -7
4. x² + 12x + 9 = 0
x² + 12x = -9
x² + 12x + 36 = -9 + 36
(x + 6)² = 27
