The problem asks to determine the range of values for the constant $a$ such that the quadratic equation $2x^2 + ax + a = 0$ has complex (imaginary) roots.

Step-by-Step Solution

1. Identify the Quadratic Equation:

   The quadratic equation is:

   $2x^2 + ax + a = 0$

2. Determine When the Roots are Complex:

   For a quadratic equation $ax^2 + bx + c = 0$, the roots are complex if the discriminant $\Delta = b^2 - 4ac$ is less than 0.

   Here, $a = 2$, $b = a$, and $c = a$.

   So, the discriminant is:

   $\Delta = a^2 - 4 \cdot 2 \cdot a = a^2 - 8a$

3. Find the Condition for Complex Roots:

   For complex roots, the discriminant must be less than 0:

   $a^2 - 8a < 0$

   Factor the quadratic inequality:

   $a(a - 8) < 0$

4. Solve the Inequality:

   To solve this inequality, find the critical points where the expression equals 0:

   $a = 0, \quad a = 8$

   Now, analyze the sign of the expression $a(a - 8)$ in the intervals defined by these points:

   • For $a < 0$, $a(a - 8) > 0$
   • For $0 < a < 8$, $a(a - 8) < 0$
   • For $a > 8$, $a(a - 8) > 0$

   The expression $a(a - 8) < 0$ when $0 < a < 8$.

5. Conclusion:

   The range of $a$ for which the quadratic equation has complex roots is:

   $\boxed{0 < a < 8}$

Answer:

The values that should be filled in the boxes are:

• For the left box (イ), the value is $0$.
• For the right box (オ), the value is $8$.

Would you like more details on any of these steps, or do you have any further questions?

Related Questions:

1. What is a discriminant and how is it used to determine the nature of roots?
2. How do you solve quadratic inequalities like $a^2 - 8a < 0$?
3. What are some examples of quadratic equations with complex roots?
4. How does the coefficient of $x^2$ affect the range of $a$ for complex roots?
5. What is the graphical representation of complex roots in a quadratic equation?

Tip:

When solving quadratic inequalities, always find the critical points and analyze the intervals they create on a number line.