Let's solve the equation:

\[
2^{6x^2} = 4^{5x+2}
\]

### Step 1: Express the equation in terms of base 2
We know that \(4 = 2^2\), so we can rewrite the equation:

\[
2^{6x^2} = (2^2)^{5x + 2}
\]

Simplifying the right-hand side:

\[
2^{6x^2} = 2^{2(5x + 2)}
\]

### Step 2: Equating exponents
Since the bases are the same (both are powers of 2), we can set the exponents equal to each other:

\[
6x^2 = 2(5x + 2)
\]

### Step 3: Solve the quadratic equation
Expand the right-hand side:

\[
6x^2 = 10x + 4
\]

Rearrange the equation to form a standard quadratic:

\[
6x^2 - 10x - 4 = 0
\]

### Step 4: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(6x^2 - 10x - 4 = 0\), we have \(a = 6\), \(b = -10\), and \(c = -4\). Plugging in these values:

\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(6)(-4)}}{2(6)}
\]

\[
x = \frac{10 \pm \sqrt{100 + 96}}{12}
\]

\[
x = \frac{10 \pm \sqrt{196}}{12}
\]

\[
x = \frac{10 \pm 14}{12}
\]

### Step 5: Find the two possible solutions
1. \(x = \frac{10 + 14}{12} = \frac{24}{12} = 2\)
2. \(x = \frac{10 - 14}{12} = \frac{-4}{12} = -\frac{1}{3}\)

### Final solutions:
The solutions are:

\[
x = 2 \quad \text{or} \quad x = -\frac{1}{3}
\]

Do you have any questions or need further explanation?

Here are 5 related questions for practice:
1. Solve \(2^{4x^2} = 8^{3x + 1}\).
2. What are the roots of the equation \(5x^2 - 7x + 2 = 0\)?
3. How do you solve the equation \(3^{2x} = 9^{x + 1}\)?
4. Simplify the expression \((2^3)^x \cdot (2^4)^{2x}\).
5. Solve the quadratic equation \(2x^2 + 3x - 5 = 0\) using the quadratic formula.

**Tip**: When solving equations involving powers, try to express them with the same base to simplify the process.

Learn how to solve the exponential equation 2^(6x^2) = 4^(5x+2) by converting it into a quadratic form and applying the quadratic formula. This solution covers exponential rules and algebraic concepts, suitable for students in Grades 10-12.

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