Question

Determine the missing values that would make this equation have zero solutions 3(5x+3) = 4x+ _________ x+9

Answer 1

To determine the values that would make the given equation \(3(5x + 3) = 4x + \_\_\_\_ x + 9\) have zero solutions, we first need to simplify and analyze the equation.

1. **Expand the left side:**
\[
3(5x + 3) = 15x + 9
\]

2. **Rewrite the right side:**
Let the missing value be \(k\). Thus, the right side becomes:
\[
4x + kx + 9 = (4 + k)x + 9
\]

3. **Now we equate both sides:**
\[
15x + 9 = (4 + k)x + 9
\]

4. **Subtract 9 from both sides:**
\[
15x = (4 + k)x
\]

5. **Rearranging gives:**
\[
15x - (4 + k)x = 0
\]
\[
(15 - (4 + k))x = 0
\]
\[
(15 - 4 - k)x = 0
\]
\[
(11 - k)x = 0
\]

To have zero solutions, we need the coefficient of \(x\) to be zero. Therefore, we set:
\[
11 - k = 0
\]

6. **Solving for \(k\):**
\[
k = 11
\]

Thus, the missing value \(k\) that would make the equation have zero solutions is:

\[
\boxed{11}
\]