Question

Z²-(10+4i)z+21+38i=0​

Answer 1

Ответ: z1 = 5+2i+3i√(2i).

z2 = 5+2i-3i√(2i).

Решение:

z²-(10+4i)z+21+38i=0​

a = 1; b = -(10+4i); c = 21+38i

D = b²-4ac

D = (-(10+4i))² - 4(21+38i) = (100+80i+16i²) -

(84+152i) = 100+80i+16i²-84-152i = 100+80i-16-84-152i = 100 - 100 - 72i = -72i

i² = -1

z1,z2 = (-b±√D)/2a

z1,z2 = ((10+4i)±√(-72i))/2 = ((10+4i)±√(72i³))/2

z1 = (10+4i+6i√(2i))/2 = 5+2i+3i√(2i).

z2 = 5+2i-3i√(2i).

Answer 2

The equation you've provided is a quadratic equation in the variable z, and it's in the form of a standard quadratic equation: az² + bz + c = 0. In your equation:

a = 1 b = -(10+4i) c = 21+38i

You can use the quadratic formula to solve for z:

z = (-b ± √(b² - 4ac)) / (2a)

Plugging in the values:

z = (-(10+4i) ± √((-10-4i)² - 4(1)(21+38i))) / (2(1))

Now, let's calculate this step by step:

1. Calculate the discriminant (the value inside the square root):

   Discriminant = (-10-4i)² - 4(1)(21+38i)

   = (100 + 80i + 16i²) - (84 + 152i)

   = (100 + 80i - 16) - (84 + 152i)

   = (100 - 16 - 84) + (80i - 152i)

   = 0 - 72i

   = -72i

2. Now, we have all the values to plug into the quadratic formula:

z = (-(10+4i) ± √(-72i)) / 2

3. Since the discriminant is -72i, the square root can be expressed as:

√(-72i) = √(72)√(-1)√i = 6√2i

4. Now, plug this into the formula:

z = (-(10+4i) ± 6√2i) / 2

5. You can factor out a common factor of 2 in the numerator:

z = -((10+4i) ± 6√2i) / 2

6. Divide each term by 2 to simplify:

z = -(5+2i) ± 3√2i

So, the two solutions for the equation Z²-(10+4i)z+21+38i=0 are:

1. z = -(5+2i) + 3√2i
2. z = -(5+2i) - 3√2i