https://file.aiquickdraw.com/mathbot/file/17302972243361q4sg2cp.jpg

The problem given is:

For which values of \( a \) does the equation
\[
9^x + 2 \cdot 3^{x+2} = a
\]
have real solutions? Solve the equation for these values of \( a \).

### Solution
Let's simplify and analyze the equation to determine the values of \( a \).

1. **Rewrite the terms in terms of \( 3^x \)**:
   \[
   9^x = (3^2)^x = (3^x)^2
   \]
   This means we can substitute \( y = 3^x \), so \( 9^x = y^2 \).

   The equation then becomes:
   \[
   y^2 + 2 \cdot y \cdot 3^2 = a
   \]

2. **Simplify further**:
   Since \( 3^2 = 9 \), we have:
   \[
   y^2 + 2 \cdot y \cdot 9 = a
   \]
   \[
   y^2 + 18y = a
   \]

3. **Determine values of \( a \) for real solutions**:
   For real values of \( x \), \( y = 3^x \) must also be real and positive (since \( 3^x \) is always positive for real \( x \)). We want to express \( a \) in terms of \( y \) and examine the resulting function:
   \[
   a = y^2 + 18y
   \]

4. **Analyze the range of \( a \)**:
   This expression \( y^2 + 18y \) is a quadratic function of \( y \) and opens upwards. The minimum value occurs at the vertex, where \( y = -\frac{18}{2} = -9 \).

   Substituting \( y = -9 \):
   \[
   a = (-9)^2 + 18 \cdot (-9) = 81 - 162 = -81
   \]

   So, the minimum value of \( a \) is \( -81 \), and \( a \) can take any value greater than or equal to \( -81 \).

### Conclusion
The equation \( 9^x + 2 \cdot 3^{x+2} = a \) has real solutions for \( a \geq -81 \).

Would you like further clarification, or do you have any specific questions?

---

### Related Questions
1. How can we find the exact values of \( x \) for specific values of \( a \)?
2. What happens to the function \( 9^x + 2 \cdot 3^{x+2} \) as \( x \to \infty \)?
3. Could we use a graphical approach to confirm the minimum value of \( a \)?
4. How does the substitution \( y = 3^x \) help in solving similar exponential equations?
5. What is the significance of the minimum value \( a = -81 \) in real-world applications?

### Tip
When dealing with exponential equations, substituting a new variable for exponential terms can greatly simplify the process.

För vilka värden på a har ekvationen 9^x + 2 * 3^(x+2) = a reella lösningar? Lös ekvationen för dessa värden på a.

Explore how to find the values of 'a' for which the equation 9^x + 2 * 3^(x+2) = a has real solutions. This problem covers exponential equations, substitution methods, and quadratic functions, ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation