https://file.aiquickdraw.com/mathbot/file/1726384584235gl9frrhk.jpg

We are given the equation:

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

The roots are specified to be non-positive and real.

Let us analyze this quadratic equation and determine the value of \( k^2 \) based on the conditions provided.

### Step 1: General Form of a Quadratic Equation
For any quadratic equation of the form:

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

The discriminant, \( \Delta \), which determines the nature of the roots, is:

\[
\Delta = b^2 - 4ac
\]

For real roots, the discriminant must satisfy:

\[
\Delta \geq 0
\]

### Step 2: Discriminant of the Given Equation
For the equation \( x^2 - ax - a^2 + 4 = 0 \), the coefficients are:

- \( b = -a \)
- \( c = -(a^2 - 4) \)

Thus, the discriminant \( \Delta \) becomes:

\[
\Delta = (-a)^2 - 4 \cdot 1 \cdot (-(a^2 - 4))
\]
\[
\Delta = a^2 + 4(a^2 - 4)
\]
\[
\Delta = a^2 + 4a^2 - 16
\]
\[
\Delta = 5a^2 - 16
\]

### Step 3: Condition for Real Roots
For real roots, we need:

\[
\Delta \geq 0
\]
\[
5a^2 - 16 \geq 0
\]
\[
5a^2 \geq 16
\]
\[
a^2 \geq \frac{16}{5}
\]
\[
a \geq \frac{4}{\sqrt{5}} \quad \text{or} \quad a \leq -\frac{4}{\sqrt{5}}
\]

### Step 4: Maximum Value of \( a \)
Since the roots are non-positive, we are interested in the negative value of \( a \). Thus, the maximum possible value of \( a \) is:

\[
a = -\frac{4}{\sqrt{5}}
\]

### Step 5: Compute \( k^2 \)
We are asked for \( k^2 \), where \( k \) is the maximum value of \( a \). Thus:

\[
k = -\frac{4}{\sqrt{5}}
\]

\[
k^2 = \left( \frac{4}{\sqrt{5}} \right)^2 = \frac{16}{5}
\]

### Final Answer:
The value of \( k^2 \) is \( \frac{16}{5} \).

Would you like further details or explanations on this solution?

---

Here are 5 related questions to extend your understanding:
1. How does the discriminant help determine the nature of the roots in a quadratic equation?
2. What conditions must be met for a quadratic equation to have non-positive real roots?
3. How do we know that the maximum value of \( a \) is negative in this case?
4. Can the quadratic equation have complex roots for different values of \( a \)?
5. What changes in the equation would make the roots positive?

**Tip:** Always check the sign of the discriminant to determine the type of roots (real or complex) before solving further!

Let the roots of x^2 - ax - a^2 + 4 = 0 be non-positive and real. If the maximum value of a is k, then k^2 is?

Learn how to solve a quadratic equation x^2 - ax - a^2 + 4 = 0 with non-positive real roots. The solution explores discriminants and finds the value of k^2. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation