It seems like you are asking how to work with the solutions derived from the quadratic formula, specifically involving complex numbers. Let’s clarify and solve it in a structured manner.

Given:

$x = \frac{-1 + \sqrt{69}i}{10} \quad \text{and} \quad x = \frac{-1 - \sqrt{69}i}{10}$

These expressions represent the solutions to the quadratic equation derived from using the quadratic formula.

To find the simplified form of these solutions:

1. Identify the quadratic formula: The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For your equation $10x^2 + 2x + 7 = 0$, we already identified $a = 10$, $b = 2$, and $c = 7$.

2. Discriminant calculation: The discriminant is: $\Delta = b^2 - 4ac = 2^2 - 4(10)(7) = 4 - 280 = -276$ Since the discriminant is negative, the solutions are complex.

3. Find the solutions: Substitute the values into the quadratic formula: $x = \frac{-2 \pm \sqrt{-276}}{20}$ The square root of a negative number gives us an imaginary number: $x = \frac{-2 \pm \sqrt{276}i}{20} = \frac{-2 \pm 2\sqrt{69}i}{20}$ Simplify further: $x = \frac{-1 \pm \sqrt{69}i}{10}$

So, the solutions you provided are indeed correct:

$x = \frac{-1 + \sqrt{69}i}{10} \quad \text{and} \quad x = \frac{-1 - \sqrt{69}i}{10}$

These are the final simplified forms using the quadratic formula.

Would you like to explore more about how these solutions relate to the graph or their properties, or do you have any other questions?

Here are five related questions:

1. What is the significance of complex roots in a quadratic equation?
2. How do complex solutions affect the graph of a quadratic equation?
3. Can complex roots be plotted on a complex plane?
4. What is the relationship between the discriminant and the nature of the roots?
5. How would you verify these solutions algebraically?

Tip: When dealing with complex numbers, remember to handle the imaginary unit $i$ carefully, especially when simplifying square roots of negative numbers.