If `4^(1+x)+4^(1-x)=10`, find x.

Answer

To solve the equation \( 4^{1+x} + 4^{1-x} = 10 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 4^{1+x} + 4^{1-x} = 10 \]
We can express \( 4^{1+x} \) and \( 4^{1-x} \) in terms of \( 4^1 \):
\[ 4 \cdot 4^x + 4 \cdot 4^{-x} = 10 \]

### Step 2: Factor out the common term
Factor out the 4 from both terms:
\[ 4(4^x + 4^{-x}) = 10 \]
Now, divide both sides by 4:
\[ 4^x + 4^{-x} = \frac{10}{4} = 2.5 \]

### Step 3: Substitute \( y = 4^x \)
Let \( y = 4^x \). Then, \( 4^{-x} = \frac{1}{y} \). Substitute these into the equation:
\[ y + \frac{1}{y} = 2.5 \]

### Step 4: Multiply through by \( y \)
To eliminate the fraction, multiply through by \( y \):
\[ y^2 + 1 = 2.5y \]

### Step 5: Rearrange into standard quadratic form
Rearranging gives us:
\[ y^2 - 2.5y + 1 = 0 \]

### Step 6: Use the quadratic formula
We can solve this quadratic equation using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -2.5 \), and \( c = 1 \):
\[ y = \frac{2.5 \pm \sqrt{(-2.5)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
Calculate the discriminant:
\[ (-2.5)^2 - 4 \cdot 1 \cdot 1 = 6.25 - 4 = 2.25 \]
Now substitute back into the formula:
\[ y = \frac{2.5 \pm \sqrt{2.25}}{2} \]
Since \( \sqrt{2.25} = 1.5 \):
\[ y = \frac{2.5 \pm 1.5}{2} \]

### Step 7: Calculate the two possible values for \( y \)
Calculating the two possible values:
1. \( y = \frac{2.5 + 1.5}{2} = \frac{4}{2} = 2 \)
2. \( y = \frac{2.5 - 1.5}{2} = \frac{1}{2} \)

### Step 8: Solve for \( x \)
Recall that \( y = 4^x \):
1. If \( 4^x = 2 \):
\[ x = \log_4(2) = \frac{1}{2} \]
2. If \( 4^x = \frac{1}{2} \):
\[ x = \log_4\left(\frac{1}{2}\right) = -\frac{1}{2} \]

### Final Answer
Thus, the possible values of \( x \) are:
\[ x = \frac{1}{2} \quad \text{or} \quad x = -\frac{1}{2} \]