help me solve an exponential equation 2^(2x+1)-5×2^x + 2 = 0
explain step by step

We are given the exponential equation:

\[
2^{(2x+1)} - 5 \cdot 2^x + 2 = 0
\]

### Step 1: Simplify the first term
The first term \(2^{(2x+1)}\) can be rewritten using properties of exponents. Specifically, we know that:

\[
2^{(2x+1)} = 2^{2x} \cdot 2^1 = 2 \cdot 2^{2x}
\]

So, the equation becomes:

\[
2 \cdot 2^{2x} - 5 \cdot 2^x + 2 = 0
\]

### Step 2: Substitution
Let’s simplify the equation further by introducing a substitution. Define:

\[
y = 2^x
\]

So, \(2^{2x} = (2^x)^2 = y^2\). Substituting this into the equation, we get:

\[
2y^2 - 5y + 2 = 0
\]

### Step 3: Solve the quadratic equation
Now, we need to solve the quadratic equation:

\[
2y^2 - 5y + 2 = 0
\]

We can use the quadratic formula:

\[
y = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2}
\]

Simplifying the terms:

\[
y = \frac{5 \pm \sqrt{25 - 16}}{4} = \frac{5 \pm \sqrt{9}}{4} = \frac{5 \pm 3}{4}
\]

This gives two possible solutions for \(y\):

\[
y_1 = \frac{5 + 3}{4} = 2 \quad \text{and} \quad y_2 = \frac{5 - 3}{4} = \frac{1}{2}
\]

### Step 4: Solve for \(x\)
Recall that \(y = 2^x\), so now we solve the equations:

- \(2^x = 2\): This gives \(x = 1\).
- \(2^x = \frac{1}{2}\): This gives \(x = -1\) (since \(2^{-1} = \frac{1}{2}\)).

### Final Answer:
The solutions to the equation are:

\[
x = 1 \quad \text{or} \quad x = -1
\]

---

Would you like more details or have any questions?

### Related Questions:
1. How would the solution change if the base was different from 2?
2. What is the geometric interpretation of solving exponential equations?
3. How does the substitution method work for more complex exponential equations?
4. What if we had an inequality instead of an equation in this problem?
5. Can you solve exponential equations graphically? How?

### Tip:
When solving exponential equations, rewriting terms with common bases and using substitution can greatly simplify the problem.

help me solve an exponential equation 2^(2x+1)-5×2^x + 2 = 0 explain step by step

Learn how to solve the exponential equation 2^(2x+1) - 5×2^x + 2 = 0 with a detailed, step-by-step breakdown. This solution covers concepts like substitution, solving quadratic equations, and exponential expressions. Suitable for students in Grades 9-12.

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