2+i+(7+i)(5-i)=

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Solution for 2+i+(7+i)(5-i)= equation:


Simplifying
2 + i + (7 + i)(5 + -1i) = 0

Multiply (7 + i) * (5 + -1i)
2 + i + (7(5 + -1i) + i(5 + -1i)) = 0
2 + i + ((5 * 7 + -1i * 7) + i(5 + -1i)) = 0
2 + i + ((35 + -7i) + i(5 + -1i)) = 0
2 + i + (35 + -7i + (5 * i + -1i * i)) = 0
2 + i + (35 + -7i + (5i + -1i2)) = 0

Combine like terms: -7i + 5i = -2i
2 + i + (35 + -2i + -1i2) = 0

Reorder the terms:
2 + 35 + i + -2i + -1i2 = 0

Combine like terms: 2 + 35 = 37
37 + i + -2i + -1i2 = 0

Combine like terms: i + -2i = -1i
37 + -1i + -1i2 = 0

Solving
37 + -1i + -1i2 = 0

Solving for variable 'i'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-37 + i + i2 = 0

Move the constant term to the right:

Add '37' to each side of the equation.
-37 + i + 37 + i2 = 0 + 37

Reorder the terms:
-37 + 37 + i + i2 = 0 + 37

Combine like terms: -37 + 37 = 0
0 + i + i2 = 0 + 37
i + i2 = 0 + 37

Combine like terms: 0 + 37 = 37
i + i2 = 37

The i term is i.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + i2 = 37 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + i2 = 37 + 0.25

Combine like terms: 37 + 0.25 = 37.25
1.25 + i2 = 37.25

Factor a perfect square on the left side:
(i + 0.5)(i + 0.5) = 37.25

Calculate the square root of the right side: 6.103277808

Break this problem into two subproblems by setting 
(i + 0.5) equal to 6.103277808 and -6.103277808.

Subproblem 1

i + 0.5 = 6.103277808 Simplifying i + 0.5 = 6.103277808 Reorder the terms: 0.5 + i = 6.103277808 Solving 0.5 + i = 6.103277808 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + i = 6.103277808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + i = 6.103277808 + -0.5 i = 6.103277808 + -0.5 Combine like terms: 6.103277808 + -0.5 = 5.603277808 i = 5.603277808 Simplifying i = 5.603277808

Subproblem 2

i + 0.5 = -6.103277808 Simplifying i + 0.5 = -6.103277808 Reorder the terms: 0.5 + i = -6.103277808 Solving 0.5 + i = -6.103277808 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + i = -6.103277808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + i = -6.103277808 + -0.5 i = -6.103277808 + -0.5 Combine like terms: -6.103277808 + -0.5 = -6.603277808 i = -6.603277808 Simplifying i = -6.603277808

Solution

The solution to the problem is based on the solutions from the subproblems. i = {5.603277808, -6.603277808}

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