Question

\\36 \times 6^{2x+6} = 6^{x^2}\\

Explanation:

Express with common base

We rewrite the equation using base \(6\).

$$ 36 \times 6^{2x+6} = 6^{x^2} $$

Since \(36 = 6^2\), we substitute this into the equation:

$$ 6^2 \times 6^{2x+6} = 6^{x^2} $$

Apply exponent rules

We combine the exponents on the left side.

$$ 6^{2 + (2x + 6)} = 6^{x^2} $$

$$ 6^{2x + 8} = 6^{x^2} $$

Equate the exponents

Since the bases are equal, their exponents must be equal.

$$ 2x + 8 = x^2 $$

Solve the quadratic equation

We rearrange the equation into standard form.

$$ x^2 - 2x - 8 = 0 $$

We factor the quadratic expression:

$$ (x - 4)(x + 2) = 0 $$

This gives two possible solutions:

$$ x = 4 \quad \text{or} \quad x = -2 $$

Answer:

\(x = 4\) or \(x = -2\)