- Simplify the equation by dividing by the leading coefficient.
- Complete the square to rewrite the equation in vertex form.
- Use the square root property to solve for \(x\), acknowledging the need for complex numbers.
- Final solution: \(\boxed{3 \pm 2i}\).
Explanation
- Simplifying and Completing the Square
We are given the equation \(8x^2 - 48x = -104\). Our goal is to solve for \(x\). We begin by simplifying the equation by dividing all terms by 8:
\[x^2 - 6x = -13\]
Next, we complete the square on the left side. The coefficient of \(x\) is \(-6\), so we add \(\left(\frac{-6}{2}\right)^2 = 9\) to both sides:
\[x^2 - 6x + 9 = -13 + 9\]
\[(x - 3)^2 = -4\]
This expression is now in a form where we can apply the square root property.
- Solving Using the Square Root Property
To solve \((x - 3)^2 = -4\), we take the square root of both sides. Since the right side is negative, the solutions will involve the imaginary unit \(i\), where \(i = \sqrt{-1}\).
Applying the square root property:
\[x - 3 = \pm \sqrt{-4}\]
\[x - 3 = \pm \sqrt{4} \cdot \sqrt{-1}\]
\[x - 3 = \pm 2i\]
Finally, we add 3 to both sides to isolate \(x\):
\[x = 3 \pm 2i\]
Thus, the solutions are \(3 + 2i\) and \(3 - 2i\). Comparing this to our options, we see that the correct choice is \(3 \pm 2i\).
Examples
Completing the square and working with complex numbers are essential skills in engineering and physics. For example, when analyzing the motion of an oscillator or the behavior of electrical circuits with alternating current, equations often take the form of quadratic expressions. When the discriminant is negative, it indicates that the system has an oscillating behavior represented by complex solutions, allowing engineers to determine the frequency and damping of the system.