$$ \Rightarrow 5 x ^ { 2 } - 45 x \quad 190 = 0 $$

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Let's first rewrite the equation clearly:

$$
5x^2 - 45x + 190 = 0
$$

Now, we can solve this quadratic equation using the quadratic formula, which is given by:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

In our equation, the coefficients are:
- $$ a = 5 $$
- $$ b = -45 $$
- $$ c = 190 $$

Now, let's calculate the discriminant $$ b^2 - 4ac $$ and then use the quadratic formula to find the values of $$ x $$.

First, we will calculate the discriminant:

$$
b^2 - 4ac = (-45)^2 - 4 \cdot 5 \cdot 190
$$

Now, I will perform the calculations for the discriminant and then use it to find the roots of the equation.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-45\right)^{2}-4\times 5\times 190$$
- step1: Multiply the terms:
  $$\left(-45\right)^{2}-3800$$
- step2: Simplify:
  $$45^{2}-3800$$
- step3: Evaluate the power:
  $$2025-3800$$
- step4: Subtract the numbers:
  $$-1775$$
The discriminant is calculated as follows:

$$
b^2 - 4ac = 2025 - 3800 = -1775
$$

Since the discriminant is negative ($$-1775$$), this means that the quadratic equation has no real solutions; instead, it has two complex solutions.

Now, we can proceed to find the complex solutions using the quadratic formula:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

Substituting the values we have:

$$
x = \frac{-(-45) \pm \sqrt{-1775}}{2 \cdot 5}
$$

This simplifies to:

$$
x = \frac{45 \pm \sqrt{-1775}}{10}
$$

Next, we can express $$\sqrt{-1775}$$ in terms of imaginary numbers:

$$
\sqrt{-1775} = i\sqrt{1775}
$$

Now, let's simplify $$\sqrt{1775}$$ to find the final solutions for $$x$$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{1775}$$
- step1: Write as a product:
  $$\sqrt{25\times 71}$$
- step2: Write in exponential form:
  $$\sqrt{5^{2}\times 71}$$
- step3: Use the properties of radicals:
  $$\sqrt{5^{2}}\times \sqrt{71}$$
- step4: Simplify the root:
  $$5\sqrt{71}$$
The square root of $$1775$$ simplifies to:

$$
\sqrt{1775} = 5\sqrt{71}
$$

Now, substituting this back into our expression for $$x$$:

$$
x = \frac{45 \pm i \cdot 5\sqrt{71}}{10}
$$

We can simplify this further:

$$
x = \frac{45}{10} \pm \frac{5i\sqrt{71}}{10}
$$

This simplifies to:

$$
x = 4.5 \pm 0.5i\sqrt{71}
$$

Thus, the two complex solutions for the equation $$5x^2 - 45x + 190 = 0$$ are:

$$
x = 4.5 + 0.5i\sqrt{71} \quad \text{and} \quad x = 4.5 - 0.5i\sqrt{71}
$$
The equation $$5x^2 - 45x + 190 = 0$$ has two complex solutions: $$x = 4.5 + 0.5i\sqrt{71}$$ and $$x = 4.5 - 0.5i\sqrt{71}$$.

\( \Rightarrow 5 x ^ { 2 } - 45 x \quad 190 = 0 \)

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