Solve $x^2-16 x+60=-12$ by completing the steps.
First, subtract □ from each side of the equation.
The given equation is \(x^2 - 16x + 60 = -12\). To solve this quadratic equation, we first aim to bring all terms to one side to set the equation to zero, or to isolate the constant term to complete the square. A logical first step is to subtract \(60\) from both sides of the equation to isolate the quadratic and linear terms:
\[x^2 - 16x = -12 - 60\]
\[x^2 - 16x = -72\]
This allows us to complete the square for the quadratic expression \(x^2 - 16x\).
To complete the square for \(x^2 - 16x\), we take the coefficient of \(x\), which is \(-16\), divide it by \(2\) to get \(-8\), and then square the result to get \((-8)^2 = 64\). We add this value, \(64\), to both sides of the equation:
\[x^2 - 16x + 64 = -72 + 64\]
This simplifies the left side into a perfect square:
\[(x - 8)^2 = -8\]
Now we solve for \(x\) by taking the square root of both sides. Since we are taking the square root of a negative number, the solutions will involve the imaginary unit \(i\), where \(i = \sqrt{-1}\):
\[x - 8 = \pm \sqrt{-8}\]
\[x - 8 = \pm \sqrt{4 \times 2 \times -1}\]
\[x - 8 = \pm 2i\sqrt{2}\]
Adding \(8\) to both sides, we find the final solutions for \(x\):
\[x = 8 \pm 2i\sqrt{2}\]
The two solutions are \(x = 8 + 2i\sqrt{2}\) and \(x = 8 - 2i\sqrt{2}\).
Quadratic equations often appear in physics when calculating the trajectory of objects or in engineering when analyzing electrical circuits. For instance, when analyzing the resonance frequency of an RLC circuit, one might encounter a characteristic equation whose solutions represent the damped oscillations of the system. Completing the square is a fundamental technique that allows us to find these critical values, helping engineers design stable and efficient systems.