Solve $\sqrt{x^4-12 x^2+36}=2(2 x+13)$ for $x$.
The given equation is \(\sqrt{x^4 - 12x^2 + 36} = 2(2x + 13)\). We start by analyzing the expression under the square root, \(x^4 - 12x^2 + 36\). This is a perfect square trinomial, as it can be rewritten in the form \((x^2)^2 - 2(6)(x^2) + 6^2\), which simplifies to \((x^2 - 6)^2\). Substituting this back into the original equation gives:
\[\sqrt{(x^2 - 6)^2} = 4x + 26\]
Recalling that \(\sqrt{a^2} = |a|\), the equation becomes:
\[|x^2 - 6| = 4x + 26\]
For this equation to have a real solution, the right side must be non-negative: \(4x + 26 \ge 0\), which implies \(x \ge -6.5\).
To solve the absolute value equation \(|x^2 - 6| = 4x + 26\), we consider two cases based on the definition of absolute value:
Case 1: \(x^2 - 6 = 4x + 26\)
Rearranging the terms, we get the quadratic equation:
\[x^2 - 4x - 32 = 0\]
This quadratic can be factored as:
\[(x - 8)(x + 4) = 0\]
This yields two potential solutions: \(x = 8\) and \(x = -4\). Since both satisfy the condition \(x \ge -6.5\), these are potential valid solutions.
Case 2: \(x^2 - 6 = -(4x + 26)\)
Rearranging the terms, we get the quadratic equation:
\[x^2 + 4x + 20 = 0\]
To find the roots, we use the discriminant \(D = b^2 - 4ac\), where \(a=1, b=4, c=20\):
\[D = 4^2 - 4(1)(20) = 16 - 80 = -64\]
Since the discriminant is negative, there are no real roots for this case.
Finally, we verify the solutions found in Case 1 in the original equation:
For \(x = 8\):
\[\sqrt{8^4 - 12(8^2) + 36} = \sqrt{4096 - 768 + 36} = \sqrt{3364} = 58\]
\[2(2(8) + 13) = 2(16 + 13) = 2(29) = 58\]
This matches.
For \(x = -4\):
\[\sqrt{(-4)^4 - 12(-4)^2 + 36} = \sqrt{256 - 192 + 36} = \sqrt{100} = 10\]
\[2(2(-4) + 13) = 2(-8 + 13) = 2(5) = 10\]
This also matches.
Comparing our results with the given options, the correct set of solutions is \(x = -4, x = 8\).
The final answer is \(\boxed{x = -4, x = 8}\).
This type of problem, involving absolute values and radical expressions, is common in physics and engineering when determining the range of a parameter, such as the position of an object or the intensity of a field, where only positive magnitudes are physically meaningful. By solving equations like this, one can accurately determine critical points where a system's behavior changes or identify valid operating conditions. For instance, if the intensity of a light source follows \(|x^2 - 6| = 4x + 26\), the identified values represent the specific positions where the light intensity satisfies the system's requirements.