%5E2%2B4%3D0.jpeg "(x-11)^2+4=0")
Solving the Quadratic Equation: (x-11)^2 + 4 = 0
This equation is a quadratic equation in the form of ax² + bx + c = 0. Let's break down the steps to solve it:
1. Simplifying the Equation
First, we need to simplify the equation by expanding the squared term:
(x - 11)² + 4 = 0 x² - 22x + 121 + 4 = 0 x² - 22x + 125 = 0
2. Using the Quadratic Formula
Now, we can use the quadratic formula to find the solutions for x:
x = (-b ± √(b² - 4ac)) / 2a
Where:
- a = 1
- b = -22
- c = 125
Plugging the values into the formula:
x = (22 ± √((-22)² - 4 * 1 * 125)) / (2 * 1) x = (22 ± √(484 - 500)) / 2 x = (22 ± √(-16)) / 2 x = (22 ± 4i) / 2
Therefore, the solutions for the equation are:
x = 11 + 2i and x = 11 - 2i
3. Understanding the Solutions
The solutions are complex numbers because the discriminant (b² - 4ac) is negative. This indicates that the quadratic equation has no real roots.
Complex numbers are numbers of the form a + bi, where:
- a is the real part
- b is the imaginary part
- i is the imaginary unit, defined as √(-1)
In this case, the solutions are 11 + 2i and 11 - 2i, both complex numbers with real part 11 and imaginary parts 2 and -2 respectively.
Conclusion
The equation (x - 11)² + 4 = 0 has two complex solutions: 11 + 2i and 11 - 2i. This demonstrates how quadratic equations can have solutions that lie beyond the realm of real numbers.