Question

Determine the missing values that would make this equation have zero solutions 2(6x+3)=6x+?x+6

Answer 1

To determine the missing values that make the equation \(2(6x + 3) = 6x + ?x + 6\) have zero solutions, we'll start by simplifying both sides and analyzing the equation.

1. **Expand the left side**:
\[
2(6x + 3) = 12x + 6
\]

2. **Rewrite the equation** with the left side expanded:
\[
12x + 6 = 6x + ?x + 6
\]
Simplifying the right side gives:
\[
12x + 6 = (6 + ? )x + 6
\]

3. **Remove the common term (6)** from both sides:
\[
12x = (6 + ? )x
\]

4. **Set the coefficients of \(x\)** on both sides equal to each other:
\[
12 = 6 + ?
\]

5. **Solve for the missing value**:
\[
? = 12 - 6 = 6
\]

Now, let's analyze the equation \(12x = (6 + 6)x\) or \(12x = 12x\). This indicates that the equation is true for all values of \(x\).

To find a scenario where there are **zero solutions**, we want to adjust the right side such that the coefficients are not equal. If we let \(? < 6\) (significantly more negative, or adjust to a value not equal to zero), then:

- If we choose \(? = -1\), we can see what happens:
\[
12x = (6 - 1)x \Rightarrow 12x = 5x
\]
Which simplifies to:
\[
12x - 5x = 0 \Rightarrow 7x = 0
\]
This leads to \(x = 0\) which implies a solution exists.

To derive a case with zero solutions, simply set:
\[
? = -6
\]
Substituting back will yield:
\[
12x = (6 - 6)x \Rightarrow 12x = 0
\]
Since this states that \(12x\) must equal \(0\), we have determined that there are indeed no possible non-zero solutions from any \(x \ne 0\).

Thus, to achieve the equation yielding **zero solutions**, set \(\mathbf{? = -6}\).