The equation 2x² + 8x + x² + 16 = 0 can be solved by first combining like terms to simplify it. So combining the x² terms gives 3x² + 8x + 16 = 0. Consider this: this is a quadratic equation in the standard form ax² + bx + c = 0, where a = 3, b = 8, and c = 16. To find the solution, we can use the quadratic formula, x = (-b ± √(b² - 4ac)) / 2a Not complicated — just consistent..
Plugging in the values, we get x = (-8 ± √(8² - 4316)) / (2*3). Simplifying further, x = (-8 ± √(64 - 192)) / 6, which becomes x = (-8 ± √(-128)) / 6. Instead, the solutions are complex numbers. Since the discriminant (b² - 4ac) is negative, the equation has no real solutions. Simplifying √(-128) gives 8i√2, so the solutions are x = (-8 ± 8i√2) / 6.
Dividing both the numerator and denominator by 2, we get x = (-4 ± 4i√2) / 3. So, the only solutions to the equation 2x² + 8x + x² + 16 = 0 are the complex numbers x = (-4 + 4i√2) / 3 and x = (-4 - 4i√2) / 3. These are the only values of x that satisfy the equation, and there are no real solutions.
Understanding the nature of the solutions is important in algebra and higher mathematics. Think about it: complex solutions often arise when the discriminant of a quadratic equation is negative, indicating that the parabola represented by the equation does not intersect the x-axis. This concept is crucial in fields such as engineering, physics, and advanced mathematics, where complex numbers are used to describe phenomena that cannot be represented by real numbers alone And that's really what it comes down to..
Not the most exciting part, but easily the most useful And that's really what it comes down to..
Simply put, the equation 2x² + 8x + x² + 16 = 0 has two complex solutions: x = (-4 + 4i√2) / 3 and x = (-4 - 4i√2) / 3. These are the only solutions, and they highlight the importance of understanding complex numbers in solving quadratic equations Surprisingly effective..
