Question

Solve for x over the complex numbers. x²+10x+41=0 Enter your answers in the boxes.

x = or x ="
a)x=5+4i,x=5−4i
b)x=5+6i,x=5−6i
c)x=10+3i,x=10−3i
d)x=3+7i,x=3−7i

Answer 1

The quadratic equation x²+10x+41=0 is solved over the complex numbers using the quadratic formula, resulting in the solutions x = 5 + 4i and x = 5 - 4i.Option A is the correct answer.

Step-by-step explanation:

To solve the quadratic equation x²+10x+41=0 for x over the complex numbers, we use the quadratic formula which is given by:

x = −b ± √(b² − 4ac) / (2a)

For the equation x²+10x+41=0, we have a = 1, b = 10, and c = 41. Substituting these values into the quadratic formula:

x = −(10) ± √((10)² − 4×41) / (2×1)

x = −10 ± √(100 − 164) / 2

x = −10 ± √(−64) / 2

x = −10 ± 8i / 2

The solutions are x = 5 + 4i and x = 5 − 4i.

To solve the quadratic equation (x² + 10x + 41 = 0) over the complex numbers, the quadratic formula is applied with coefficients (a = 1), (b = 10), and (c = 41). Substituting these values, the solutions are found to be (x = 5 + 4i) and (x = 5 - 4i). These complex solutions arise due to the expression under the square root being negative, indicating the presence of imaginary numbers. Thus, the quadratic formula provides a comprehensive method for determining solutions to quadratic equations, including those with complex roots.