$$ \left. \begin{array} { l } { ( x - 4 ) ^ { 2 } = } \ { ( 2 x + 3 ) ^ { 2 } = } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { ( x - 4 ) ^ { 2 } = } \ { ( 2 x + 3 ) ^ { 2 } = } \end{array} ight. $$

UpStudy AI · Accepted Answer

Solve the system of equations by following steps:
- step0: Solution:
  $$\left\{ \begin{array}{l}\left(x-4\right)^{2}=0\\left(2x+3\right)^{2}=0\end{array}\right.$$
- step1: Calculate:
  $$\left\{ \begin{array}{l}x=4\x=-\frac{3}{2}\end{array}\right.$$
- step2: Find the intersection:
  $$x \in \varnothing $$
The system of equations given is:

1. $$ (x - 4)^{2} = 0 $$
2. $$ (2x + 3)^{2} = 0 $$

Let's analyze each equation step by step.

### Step 1: Solve the first equation
The first equation can be simplified as follows:

$$
(x - 4)^{2} = 0
$$

Taking the square root of both sides gives:

$$
x - 4 = 0
$$

Thus, we find:

$$
x = 4
$$

### Step 2: Solve the second equation
Now, let's solve the second equation:

$$
(2x + 3)^{2} = 0
$$

Taking the square root of both sides gives:

$$
2x + 3 = 0
$$

Solving for $$x$$:

$$
2x = -3 \implies x = -\frac{3}{2}
$$

### Step 3: Combine the results
The solutions we found are:

- From the first equation: $$ x = 4 $$
- From the second equation: $$ x = -\frac{3}{2} $$

However, since these two values of $$x$$ do not satisfy both equations simultaneously, the system has no common solution.

### Conclusion
The result of the system of equations is:

$$
x \in \varnothing
$$

This means there are no values of $$x$$ that satisfy both equations at the same time.
There are no solutions for $$x$$ that satisfy both equations simultaneously.

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