Solve for x 10^(2x)+11=(x+6)^2-2

$10^{2 x} + 11 = {(x + 6)}^{2} - 2$

Simplify ${(x + 6)}^{2} - 2$ .

Simplify each term.

Rewrite ${(x + 6)}^{2}$ as $(x + 6) (x + 6)$ .

$10^{2 x} + 11 = (x + 6) (x + 6) - 2$

Expand $(x + 6) (x + 6)$ using the FOIL Method.

Apply the distributive property.

$10^{2 x} + 11 = x (x + 6) + 6 (x + 6) - 2$

Apply the distributive property.

$10^{2 x} + 11 = x \cdot x + x \cdot 6 + 6 (x + 6) - 2$

Apply the distributive property.

$10^{2 x} + 11 = x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6 - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$10^{2 x} + 11 = x^{2} + x \cdot 6 + 6 x + 6 \cdot 6 - 2$

Move $6$ to the left of $x$ .

$10^{2 x} + 11 = x^{2} + 6 \cdot x + 6 x + 6 \cdot 6 - 2$

Multiply $6$ by $6$ .

$10^{2 x} + 11 = x^{2} + 6 x + 6 x + 36 - 2$

Add $6 x$ and $6 x$ .

$10^{2 x} + 11 = x^{2} + 12 x + 36 - 2$

Subtract $2$ from $36$ .

$10^{2 x} + 11 = x^{2} + 12 x + 34$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 9.60555127, - 2.39444646, 0.75698733$