Atynice the Jakes of the connery $p$ and $Q$ equation.
$x^{2 x}-Q^{2 x}=6 \operatorname{arch} 2 x+4 \cos 12$
Answer (2)
To find Q in terms of x from the given equation, we isolate Q 2 x and take the ( 2 x ) -th root. The final expression for Q is Q = ( x 2 x − 6 arch ( 2 x ) − 4 cos ( 12 ) ) 2 x 1 .
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Isolate Q 2 x on one side of the equation.
Take the ( 2 x ) -th root of both sides to solve for Q .
The final expression for Q in terms of x is: Q = ( x 2 x − 6 arch 2 x − 4 cos 12 ) 2 x 1 .
Explanation
Understanding the Problem The problem asks us to find Q in terms of x given the equation x 2 x − Q 2 x = 6 arch 2 x + 4 cos 12
Isolating Q First, we isolate the term containing Q :
Q 2 x = x 2 x − 6 arch 2 x − 4 cos 12
Solving for Q Next, we take the ( 2 x ) -th root of both sides to solve for Q :
Q = ( x 2 x − 6 arch 2 x − 4 cos 12 ) 2 x 1 Note that arch ( u ) is the inverse hyperbolic cosine function, often written as cosh − 1 ( u ) .
Final Expression for Q Therefore, the expression for Q in terms of x is: Q = 2 x x 2 x − 6 arch ( 2 x ) − 4 cos ( 12 )
Examples
In physics, equations relating variables are common. For instance, relating the energy E of a particle to its momentum p and mass m via E 2 = p 2 c 2 + m 2 c 4 , where c is the speed of light. Solving for one variable in terms of others allows physicists to predict how one quantity changes with respect to others, which is crucial in experimental design and theoretical understanding. Similarly, in electrical engineering, one might solve for the impedance of a circuit in terms of resistance, inductance, and capacitance to optimize circuit performance.
