- Identify the coefficients \(a = -10, b = 12, c = -9\) for the quadratic equation \(-10x^2 + 12x - 9 = 0\).
- Calculate the discriminant \(D = b^2 - 4ac = -216\), which leads to complex roots \(\sqrt{D} = 6i\sqrt{6}\).
- Use the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) to obtain the roots.
- Simplify the expression to find \(x = \frac{3}{5} \pm \frac{3i\sqrt{6}}{10}\), which corresponds to the first option. \(\boxed{x = \frac{3}{5} \pm \frac{3i\sqrt{6}}{10}}\)
Explanation
- Problem Analysis and Setup
The given equation is a quadratic equation in the standard form \(ax^2 + bx + c = 0\), specifically:
\[-10x^2 + 12x - 9 = 0.\]
To solve for the roots, we identify the coefficients:
\(a = -10\),
\(b = 12\),
\(c = -9\).
The quadratic formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]
We will substitute these values into the formula to find the roots.
- Calculating the Discriminant and Square Root
First, calculate the discriminant \(D = b^2 - 4ac\):
\[D = (12)^2 - 4(-10)(-9) = 144 - 360 = -216.\]
Since the discriminant is negative, the roots will be complex conjugates. We can write \(\sqrt{D}\) as \(\sqrt{-216} = \sqrt{216} \times i\).
Since \(216 = 36 \times 6\), we have:
\[\sqrt{216} = \sqrt{36} \times \sqrt{6} = 6\sqrt{6}.\]
Therefore, \(\sqrt{D} = 6i\sqrt{6}\).
- Final Calculation and Root Simplification
Now, substitute the discriminant back into the quadratic formula:
\[x = \frac{-12 \pm 6i\sqrt{6}}{2(-10)} = \frac{-12 \pm 6i\sqrt{6}}{-20}.\]
Divide both parts of the numerator by the denominator \(-20\):
\[x = \frac{-12}{-20} \pm \frac{6i\sqrt{6}}{-20}.\]
Simplifying the fractions:
\(\frac{-12}{-20} = \frac{3}{5}\)
\(\frac{6i\sqrt{6}}{-20} = -\frac{3i\sqrt{6}}{10}\) (or simply using the \(\pm\) sign, it becomes \(\mp \frac{3i\sqrt{6}}{10}\)).
Since \(\pm\) covers both cases, the roots are:
\[x = \frac{3}{5} \pm \frac{3i\sqrt{6}}{10}.\]
This matches the first option.
Examples
Quadratic equations with complex roots are frequently encountered in electrical engineering when analyzing RLC circuits (Resistors, Inductors, and Capacitors). The complex components of the solution represent the oscillation frequency and decay rate of the circuit's response. Understanding how to find these roots allows engineers to design systems that are stable and behave according to specific requirements, ensuring reliable operation in various electronic devices.