- Identify the coefficients \(a = -5, b = -5, c = -8\) from the equation \(-5k^2 - 5k - 8 = 0\).
- Calculate the discriminant \(D = b^2 - 4ac = -135\).
- Apply the quadratic formula \(k = \frac{-b \pm \sqrt{D}}{2a}\) to find the solutions.
- Simplify to reach the final solutions: \(k = -\frac{1}{2} \pm \frac{3i\sqrt{15}}{10}\).
- The solutions are \(k = \boxed{-\frac{1}{2} + \frac{3i\sqrt{15}}{10}}\) and \(k = \boxed{-\frac{1}{2} - \frac{3i\sqrt{15}}{10}}\).
Explanation
- Identifying Coefficients
To solve the quadratic equation \(-5k^2 - 5k - 8 = 0\), we identify the coefficients \(a, b,\) and \(c\). The standard form of a quadratic equation is \(ak^2 + bk + c = 0\). From the given equation, we have \(a = -5\), \(b = -5\), and \(c = -8\).
- Calculating the Discriminant
The quadratic formula for finding the roots of a quadratic equation is given by:
\[k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
We first calculate the discriminant \(D = b^2 - 4ac\):
\[D = (-5)^2 - 4 \times (-5) \times (-8)\]
\[D = 25 - 160 = -135\]
Since the discriminant is negative, the solutions for \(k\) will be complex numbers.
- Applying the Quadratic Formula
Now we substitute the values of \(a, b, c\), and the discriminant into the quadratic formula:
\[k = \frac{-(-5) \pm \sqrt{-135}}{2 \times (-5)}\]
\[k = \frac{5 \pm \sqrt{-135}}{-10}\]
We can simplify the term \(\sqrt{-135}\) as \(i\sqrt{135}\). Since \(135 = 9 \times 15\), we have \(\sqrt{135} = \sqrt{9} \times \sqrt{15} = 3\sqrt{15}\).
Therefore, \(\sqrt{-135} = 3i\sqrt{15}\).
Substituting this back, we get:
\[k = \frac{5 \pm 3i\sqrt{15}}{-10}\]
- Simplifying the Results
We can further simplify the expression by splitting the fraction:
\[k = \frac{5}{-10} \pm \frac{3i\sqrt{15}}{-10}\]
\[k = -\frac{1}{2} \mp \frac{3i\sqrt{15}}{10}\]
This gives us the two complex solutions:
\[k = -\frac{1}{2} + \frac{3i\sqrt{15}}{10} \quad \text{and} \quad k = -\frac{1}{2} - \frac{3i\sqrt{15}}{10}\]
Examples
The quadratic formula is essential for finding the roots of parabolic functions, which appear frequently in physics and engineering. For example, if you are modeling the trajectory of a projectile and the equation for its height results in a negative discriminant, it indicates that the object never reaches the ground at that specific elevation or speed, or that the path is purely imaginary in your coordinate system. Understanding complex roots helps engineers and physicists analyze systems where oscillations or damping occur, ensuring safe and effective designs in structures, electronics, and beyond.