We are tasked with solving for $x$ in the equation:

$x = 4 \left[ \frac{68 + j70 - j36 \tan(135^\circ)}{52 - 3(68 + j70) \tan(135^\circ)} \right]$

Let's break it down step by step.

Step 1: Analyze the components inside the equation

The equation involves a complex number notation, with $j$ representing the imaginary unit ($j = \sqrt{-1}$). We need to simplify this expression by evaluating the trigonometric part and simplifying the real and imaginary parts.

Step 2: Evaluate $\tan(135^\circ)$

We know that: $\tan(135^\circ) = -1$

Step 3: Substitute $\tan(135^\circ) = -1$ into the equation

Substituting into the original equation:

$x = 4 \left[ \frac{68 + j70 - j36 (-1)}{52 - 3(68 + j70)(-1)} \right]$

This simplifies further to:

$x = 4 \left[ \frac{68 + j70 + j36}{52 + 3(68 + j70)} \right]$

Step 4: Simplify the numerator and denominator

Numerator:

$68 + j70 + j36 = 68 + j(70 + 36) = 68 + j106$

Denominator:

$52 + 3(68 + j70) = 52 + 3(68) + 3(j70) = 52 + 204 + j210 = 256 + j210$

Step 5: Substitute these simplified forms back into the equation for $x$

Now the equation for $x$ becomes:

$x = 4 \left[ \frac{68 + j106}{256 + j210} \right]$

Step 6: Simplify the complex fraction

We need to simplify the expression:

$x = 4 \left[ \frac{68 + j106}{256 + j210} \right]$

To simplify a complex fraction, we multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of $256 + j210$ is $256 - j210$.

Multiply both numerator and denominator by $256 - j210$:

$x = 4 \left[ \frac{(68 + j106)(256 - j210)}{(256 + j210)(256 - j210)} \right]$

Denominator:

$(256 + j210)(256 - j210) = 256^2 - (j210)^2 = 65536 - (-44100) = 65536 + 44100 = 109636$

Numerator:

Now, expand the numerator:

$(68 + j106)(256 - j210) = 68 \cdot 256 + 68 \cdot (-j210) + j106 \cdot 256 + j106 \cdot (-j210)$

$= 17408 - j14280 + j27136 - 22260$

Combine like terms:

$= (17408 - 22260) + j(-14280 + 27136)$

$= -4852 + j12856$

Step 7: Final equation

Thus, we have:

$x = 4 \left[ \frac{-4852 + j12856}{109636} \right]$

This simplifies to:

$x = \frac{4(-4852 + j12856)}{109636}$

Breaking it into real and imaginary components:

$x = \frac{-19408 + j51424}{109636}$

Now, simplify each part:

$x = \frac{-19408}{109636} + j \frac{51424}{109636}$

Final Answer:

$x = -0.177 + j 0.469$

Thus, the solution for $x$ is:

$x \approx -0.177 + j0.469$

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Would you like further explanation on any part of this process? Or perhaps explore another similar problem?

Here are 5 related questions you might find interesting:

1. How can you simplify a complex number division without using the conjugate?
2. How do you calculate the tangent of angles other than 45° and 135°?
3. What does the real and imaginary part represent in a complex solution?
4. Can this process be used for more complicated complex number equations?
5. How can you plot complex numbers on the complex plane?

Tip: When multiplying complex numbers, remember to use the distributive property to expand the terms and then combine like terms.