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The given quadratic equation is:

\[
x^2 - ax - a^2 + 4 = 0
\]

It is mentioned that the roots are non-positive and real. You are asked to find the minimum value of \( a \), which is denoted by \( \lambda \), and then determine \( \lambda^2 \).

### Step-by-step solution:

1. **Form of the Quadratic Equation:**

   The equation is:

   \[
   x^2 - ax - a^2 + 4 = 0
   \]

   The general form of a quadratic equation is:

   \[
   x^2 + bx + c = 0
   \]

   where \( b = -a \) and \( c = -a^2 + 4 \).

2. **Sum and Product of Roots:**

   For a quadratic equation \( x^2 + bx + c = 0 \), the sum and product of the roots are given by:

   \[
   \text{Sum of roots} = -\frac{b}{1} = a
   \]
   \[
   \text{Product of roots} = \frac{c}{1} = -a^2 + 4
   \]

   Let the roots of the equation be \( r_1 \) and \( r_2 \). From Vieta's formulas:

   \[
   r_1 + r_2 = a
   \]
   \[
   r_1 \cdot r_2 = -a^2 + 4
   \]

3. **Condition for Non-Positive Roots:**

   It is given that the roots are non-positive and real. This implies:
   
   - Both roots are either negative or zero.
   - The sum of the roots \( r_1 + r_2 \) should be non-positive, meaning \( a \leq 0 \).
   - The product of the roots \( r_1 \cdot r_2 \geq 0 \) because the product of two non-positive numbers is non-negative.

   So, we have the following conditions:

   - \( a \leq 0 \)
   - \( -a^2 + 4 \geq 0 \)

4. **Finding the Minimum Value of \( a \):**

   The second condition is \( -a^2 + 4 \geq 0 \), or equivalently:

   \[
   a^2 \leq 4
   \]

   This gives:

   \[
   -2 \leq a \leq 2
   \]

   Since the roots are non-positive, we must have \( a \leq 0 \). Therefore, the range of \( a \) is \( -2 \leq a \leq 0 \).

   The minimum value of \( a \) is \( \lambda = -2 \).

5. **Calculating \( \lambda^2 \):**

   The minimum value of \( a \) is \( \lambda = -2 \), so:

   \[
   \lambda^2 = (-2)^2 = 4
   \]

Thus, the value of \( \lambda^2 \) is **4**.

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

Here are five related questions for practice:
1. How do you derive the discriminant condition for real roots of a quadratic equation?
2. What does Vieta's formula say about the sum and product of roots in a quadratic equation?
3. If the roots of a quadratic equation are negative, what does it imply about the sign of the discriminant?
4. How would the result change if the roots were positive instead of non-positive?
5. What are the general steps for finding the minimum or maximum value of a quadratic function?

**Tip:** When dealing with quadratic equations, always check whether conditions on the roots (like positivity or negativity) affect the sum and product relationships.

Let roots of x^2 - ax - a^2 + 4 = 0 are non-positive and real. If the minimum value of a is λ, then λ^2 is

Learn how to determine the minimum value of a for a quadratic equation with non-positive real roots, using algebraic techniques, Vieta's formulas, and inequalities. Suitable for students in Grades 9-11.

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