Question

Option #1: 3(2x+5)=6x+15 Option #2: 8x+7=5x+3x+1+6 Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24 Option #4: 2x12⋅3=6x−−√

Answer 1

I got the answer wrong. It's 3

Answer 2

Let's solve each of the options step by step:

**Option #1: \(3(2x+5) = 6x + 15\)**

Distributing on the left side:
\[ 3 \cdot 2x + 3 \cdot 5 = 6x + 15 \]
\[ 6x + 15 = 6x + 15 \]

This equation is true for all values of \(x\). Thus, Option #1 is an identity.

---

**Option #2: \(8x+7=5x+3x+1+6\)**

First, simplify the right side:
\[ 8x + 7 = (5x + 3x) + 1 + 6 \]
\[ 8x + 7 = 8x + 7 \]

This equation is also true for all values of \(x\). Thus, Option #2 is an identity.

---

**Option #3: \((2x^2 − 4)(3y + 6) = 6x^2y + 12x^2 − 12y − 24\)**

First, let's check the left-hand side.
\[ (2x^2 - 4)(3y + 6) \]

Distributing:
\[ 2x^2 \cdot 3y + 2x^2 \cdot 6 - 4 \cdot 3y - 4 \cdot 6 \]
\[ = 6x^2y + 12x^2 - 12y - 24 \]

The left-hand side simplifies to the same as the right-hand side:
\[ 6x^2y + 12x^2 - 12y - 24 = 6x^2y + 12x^2 - 12y - 24 \]

This equation holds true, so Option #3 is also an identity.

---

**Option #4: \(2x^{12} \cdot 3 = 6x^{\frac{1}{2}}\)**

First, simplify the left-hand side:
\[ 2x^{12} \cdot 3 = 6x^{12} \]

Now, we compare it with the right-hand side:
\[ 6x^{12} = 6x^{\frac{1}{2}} \]

Dividing both sides by 6 gives:
\[ x^{12} = x^{\frac{1}{2}} \]

This only holds true if \(x \neq 0\) and under certain conditions. Specifically:
\[ 12 = \frac{1}{2} \Rightarrow 24 = 1 \]
which is not true.

Thus, Option #4 does not hold true for all \(x\) values.

---

### Conclusion:
- Option #1: Identity
- Option #2: Identity
- Option #3: Identity
- Option #4: Not valid for all \(x\)

The only option that is not valid universally is Option #4.

Answer 3

Apologies for the confusion. Let's take a closer look at Option #3:

**Option #3: \((2x^2−4)(3y+6)=6x^2y+12x^2−12y−24\)**

Let's simplify the left-hand side step by step.

1. **Expand the Left Side:**
\[
(2x^2 - 4)(3y + 6)
\]
Using the distributive property:
\[
= 2x^2(3y) + 2x^2(6) - 4(3y) - 4(6)
\]
\[
= 6x^2y + 12x^2 - 12y - 24
\]

2. **Set the Left Side Equal to the Right Side:**
\[
6x^2y + 12x^2 - 12y - 24 = 6x^2y + 12x^2 - 12y - 24
\]

Here, after performing the expansion, we see that both sides are indeed equal.

So, you were correct in stating that Option #3 is **true**, meaning it holds for all \(x\) and \(y\).

### Summary of the Results:
- **Option #1**: Identity
- **Option #2**: Identity
- **Option #3**: Identity (holds true)
- **Option #4**: Not valid for all \(x\)

If the answer you were looking for was indeed Option #3 as the one that holds true universally, then you are right. Thank you for your patience, and let me know if you need further clarification!