I really need help factoring these special Trinomials.
This question really puzzled me. I thought I had it correct, but when I checked the answer at the back of the book, I was wrong, and so I would like to know what was it that I had done wrong.
This was the question:
x^2-4x+4
This is how I solved it:
x^2-4x+4
x^2-2x-2x+4
x(x-2)-2(x+2)
(x-2)(x+2)
- I could not get any further than this. I thought that this would be correct, but the actual answer is
(x-2)^2
How did they get that?
If there is an easier way to solve this, then can you please show me how. Cause solving things like I have above is difficult.
Your error is in this line of yours
x(x-2)-2(x+2)
notice when you multiply it back out you don't get your original, and you don't have a "common" factor.
should have been
x(x-2)-2(x-2) to get back your x^2 - 4x + 4
= (x-2)(x-2)
= (x-2)^2
To factor the special trinomial x^2 - 4x + 4 correctly, we can use a technique called "completing the square."
Here's how you can factor it correctly:
Step 1: Write down the trinomial expression: x^2 - 4x + 4.
Step 2: Look at the first term, x^2. Take its square root, which is x.
Step 3: Look at the last term, 4. Take its square root, which is 2.
Step 4: Split the middle term -4x into two terms using the values obtained in Steps 2 and 3. In this case, -4x can be split as -2x - 2x.
Step 5: Rewrite the trinomial by replacing the original middle term with the newly split terms:
x^2 - 2x - 2x + 4.
Step 6: Group the terms in pairs:
(x^2 - 2x) - (2x - 4).
Step 7: Factor out the greatest common factor (GCF) from each pair of terms:
x(x - 2) - 2(x - 2).
Step 8: Notice that (x - 2) is a common factor, so you can extract it:
(x - 2)(x - 2).
Step 9: Simplify by writing (x - 2) as (x - 2)^2:
(x - 2)^2.
So, the correct factored form is (x - 2)^2.
Completing the square may seem a bit complicated at first, but with practice, it becomes easier. It is a useful technique for factoring special trinomials.