To solve the problem of simplifying the expression [tex]\(3x(x - 12x) + 3x^2 - 2(x - 2)^2\)[/tex], let’s go through each step thoroughly:
1. Expanding the Expression:
- First, simplify the term [tex]\(3x(x - 12x)\)[/tex]:
[tex]\[
3x(x - 12x) = 3x \cdot (-11x) = 3x \cdot (-11x) = -33x^2
\][/tex]
- Now we have:
[tex]\[
-33x^2 + 3x^2 - 2(x - 2)^2
\][/tex]
2. Simplifying the Square Term:
- Next, simplify the term [tex]\(-2(x - 2)^2\)[/tex]:
[tex]\[
(x - 2)^2 = x^2 - 4x + 4
\][/tex]
[tex]\[
-2(x^2 - 4x + 4) = -2x^2 + 8x - 8
\][/tex]
- Now our expression is:
[tex]\[
-33x^2 + 3x^2 - 2x^2 + 8x - 8
\][/tex]
3. Combining Like Terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-33x^2 + 3x^2 - 2x^2 = -33x^2 + 3x^2 - 2x^2 = -32x^2
\][/tex]
- The expression now looks like:
[tex]\[
-32x^2 + 8x - 8
\][/tex]
4. Final Simplified Expression:
- Hence, the simplified expression is:
[tex]\[
-32x^2 + 8x - 8
\][/tex]
Now let's examine the given statements:
1. The term [tex]\(-2(x-2)^2\)[/tex] is simplified by first squaring the expression [tex]\(x-2\)[/tex].
- Yes, this is true. We squared the term inside the parentheses first, and then we distributed the [tex]\(-2\)[/tex].
2. The simplified product is a binomial.
- No, this is false. A binomial has exactly two terms. The final simplified expression has three terms: [tex]\(-32x^2 + 8x - 8\)[/tex].
3. After multiplying, the like terms are combined by adding and subtracting.
- This is true. We added and subtracted the coefficients of the [tex]\(x^2\)[/tex] terms while simplifying.
4. The parentheses are eliminated through multiplication.
- This is also true. In the process of expansion, we multiplied out the terms to eliminate the parentheses.
5. The final simplified product is [tex]\(-28x^2 + 8x - 8\)[/tex].
- This is false. The correct simplified expression is [tex]\(-32x^2 + 8x - 8\)[/tex].
So, the correct statements are:
- The term [tex]\(-2(x-2)^2\)[/tex] is simplified by first squaring the expression [tex]\(x-2\)[/tex].
- After multiplying, the like terms are combined by adding and subtracting.
- The parentheses are eliminated through multiplication.
