Rewrite the trinomial -x^2 + 8x - 16 as the square of a binomial or as the opposite of the square of a binomial.
WRONG AGAIN, expand the bot's answer to see that you don't get the original.
-x^2 + 8x - 16
= -(x^2 - 8x + 16)
= - (x - 4)^2
To rewrite the trinomial -x^2 + 8x - 16 as the square of a binomial or the opposite of the square of a binomial, we need to use the formula for expanding a perfect square trinomial.
The formula for expanding a perfect square trinomial is:
(a + b)^2 = a^2 + 2ab + b^2
In this case, we can rewrite the trinomial by completing the square. Let's start with the first term -x^2:
(-x)^2 = x^2
Next, we need to find the middle term's coefficient, which is 8, and divide it by 2:
8/2 = 4
So, the middle term is ny 4x. We add and subtract 4x from the trinomial:
-x^2 + 4x - 4x + 8x - 16
Now, we group the terms:
((-x^2 + 4x) - 4x) + (8x - 16)
Next, we factor out a common factor from each group:
x(-x + 4) - 4(-x + 4)
Then, we combine the common factors:
(x - 4)(-x + 4)
Finally, we can rewrite the trinomial -x^2 + 8x - 16 as the difference of two squares:
-(x - 4)^2
So, -x^2 + 8x - 16 can be rewritten as the negative square of the binomial (x - 4).
To rewrite the trinomial -x^2 + 8x - 16 as the square of a binomial or as the opposite of the square of a binomial, we need to complete the square. The general form of a quadratic trinomial in the form of x^2 + bx + c can be factored as (x + p)^2 + q, where p is half the coefficient of x and q is the constant term.
Step 1: Group the terms
Start by grouping the terms of the trinomial: -x^2 + 8x - 16.
Step 2: Factor out the common negative sign
Factor out a -1 from -x^2 and 8x to simplify the expression: -(x^2 - 8x) - 16.
Step 3: Complete the square by adding the square of half the coefficient of x
To complete the square, we need to add the square of half the coefficient of x, which is (8/2)^2 = 16.
-(x^2 - 8x + 16) - 16 - 16 => -(x^2 - 8x + 16) - 32
Step 4: Rewrite as the square of a binomial or the opposite of the square of a binomial
Factor the perfect square trinomial (x - 4)^2 = x^2 - 8x + 16 and rewrite the expression:
-(x - 4)^2 - 32
So, the trinomial -x^2 + 8x - 16 can be rewritten as the opposite of the square of the binomial x - 4, or as -(x - 4)^2, minus 32.
We can start by looking at the first two terms (-x^2 + 8x) since a perfect square trinomial always has the form of (a + b)^2 = a^2 + 2ab + b^2, where a and b are constants.
We can see that there is a common factor of -x in the first two terms, so we can rewrite the trinomial as:
-x^2 + 8x - 16 = -x(x - 8) - 16
Now, we can see that we have two terms: -x and (x - 8), which we can try to rewrite in the form of (a + b)^2 or (a - b)^2.
For example, if we take -x = -1x and (x - 8) = (-1(-8)), we can rewrite the trinomial as:
-x^2 + 8x - 16 = (-1x + (-8))^2 - 16
Therefore, we can rewrite the trinomial as the square of a binomial with -1x - 8 as its factor:
-x^2 + 8x - 16 = (-1x - 8)^2.
Alternatively, we could also write:
-x^2 + 8x - 16 = -((-1x + 4)^2 - 0)
So, in this case, the trinomial can be rewritten as the opposite of the square of a binomial, which is (-1x + 4)^2 with the opposite sign.