$$ 2.2 \quad 4 ^ { x } - 2 ^ { x } = 2 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
4^{x} - 2^{x} = 2
$$

we can follow these steps:

### 1. Make a Substitution

Let’s set $$ y = 2^{x} $$. Notice that $$ 4^{x} $$ can be rewritten in terms of $$ y $$:

$$
4^{x} = (2^{2})^{x} = 2^{2x} = (2^{x})^{2} = y^{2}
$$

### 2. Rewrite the Equation

Substitute $$ y $$ into the original equation:

$$
y^{2} - y = 2
$$

### 3. Form a Quadratic Equation

Bring all terms to one side to form a quadratic equation:

$$
y^{2} - y - 2 = 0
$$

### 4. Solve the Quadratic Equation

Use the quadratic formula to solve for $$ y $$:

$$
y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
$$

For the equation $$ y^{2} - y - 2 = 0 $$, the coefficients are $$ a = 1 $$, $$ b = -1 $$, and $$ c = -2 $$. Plugging these into the formula:

$$
y = \frac{-(-1) \pm \sqrt{(-1)^{2} - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 + 8}}{2} = \frac{1 \pm 3}{2}
$$

This yields two solutions:

$$
y = \frac{1 + 3}{2} = 2 \quad \text{and} \quad y = \frac{1 - 3}{2} = -1
$$

### 5. Consider Valid Solutions

Since $$ y = 2^{x} $$ and $$ 2^{x} > 0 $$ for all real $$ x $$, the solution $$ y = -1 $$ is not valid. Therefore, we only consider:

$$
y = 2
$$

### 6. Solve for $$ x $$

Recall that $$ y = 2^{x} $$:

$$
2^{x} = 2
$$

Taking the logarithm base 2 of both sides:

$$
x = \log_{2}(2) = 1
$$

### 7. Verify the Solution

Plug $$ x = 1 $$ back into the original equation to verify:

$$
4^{1} - 2^{1} = 4 - 2 = 2
$$

Which matches the right-hand side of the equation.

### **Conclusion**

The solution to the equation $$ 4^{x} - 2^{x} = 2 $$ is:

$$
\boxed{1}
$$
$$ x = 1 $$

Upstudy AI Solution