Complete set of values of k for which the equation 4^x – (k + 2)2^x + 2k = 0, has exactly one positive root, is

We are given the equation \( 4^x - (k+2)2^x + 2k = 0 \), and we need to find the complete set of values of \( k \) for which this equation has exactly one positive root.

### Step 1: Simplify the equation
We know that \( 4^x = (2^x)^2 \). Let’s substitute \( 2^x = t \), where \( t > 0 \) because \( 2^x \) is always positive for any real \( x \). The equation becomes:

\[
t^2 - (k+2)t + 2k = 0
\]

This is a quadratic equation in \( t \). We now want this quadratic equation to have exactly **one positive solution** for \( t \), which corresponds to having exactly one positive root for \( x \).

### Step 2: Condition for one positive root
For a quadratic equation to have exactly one root, its discriminant must be zero. The discriminant \( \Delta \) of the quadratic equation \( at^2 + bt + c = 0 \) is given by:

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

In our case, \( a = 1 \), \( b = -(k+2) \), and \( c = 2k \). So, the discriminant becomes:

\[
\Delta = (-(k+2))^2 - 4(1)(2k) = (k+2)^2 - 8k
\]

Expanding this:

\[
\Delta = k^2 + 4k + 4 - 8k = k^2 - 4k + 4
\]

For the quadratic equation to have exactly one solution, we set the discriminant equal to zero:

\[
k^2 - 4k + 4 = 0
\]

### Step 3: Solve for \( k \)
The equation \( k^2 - 4k + 4 = 0 \) is a perfect square trinomial:

\[
(k - 2)^2 = 0
\]

Solving this, we get:

\[
k = 2
\]

### Step 4: Verify the solution
When \( k = 2 \), the original quadratic equation becomes:

\[
t^2 - (2+2)t + 2(2) = t^2 - 4t + 4 = 0
\]

This simplifies to:

\[
(t - 2)^2 = 0
\]

Thus, \( t = 2 \), which means \( 2^x = 2 \), and hence \( x = 1 \).

Since there is exactly one positive root, the value of \( k \) is indeed \( 2 \).

### Final Answer:
The complete set of values of \( k \) for which the equation has exactly one positive root is:

\[
\boxed{2}
\]

---

Would you like more details on any step? 

Here are five related questions:
1. How does the discriminant of a quadratic equation determine the number of roots?
2. What is the significance of the substitution \( 2^x = t \) in solving the equation?
3. What happens if \( k \neq 2 \)? Can the equation have more than one root?
4. Can this method be applied to other equations with similar forms?
5. What are the possible types of roots (real or complex) for quadratic equations?

**Tip**: Always check the discriminant when solving quadratic equations, as it tells you how many real solutions exist!

Explore how to find the complete set of values of k for the equation 4^x - (k + 2)2^x + 2k = 0, which has exactly one positive root. The solution uses algebraic techniques, quadratic equations, and discriminants, suitable for 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