3i(7+2i)-(5+4i)=

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Solution for 3i(7+2i)-(5+4i)= equation:


Simplifying
3i(7 + 2i) + -1(5 + 4i) = 0
(7 * 3i + 2i * 3i) + -1(5 + 4i) = 0
(21i + 6i2) + -1(5 + 4i) = 0
21i + 6i2 + (5 * -1 + 4i * -1) = 0
21i + 6i2 + (-5 + -4i) = 0

Reorder the terms:
-5 + 21i + -4i + 6i2 = 0

Combine like terms: 21i + -4i = 17i
-5 + 17i + 6i2 = 0

Solving
-5 + 17i + 6i2 = 0

Solving for variable 'i'.

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

Divide each side by '6'.
-0.8333333333 + 2.833333333i + i2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + 2.833333333i + 0.8333333333 + i2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + 2.833333333i + i2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + 2.833333333i + i2 = 0 + 0.8333333333
2.833333333i + i2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
2.833333333i + i2 = 0.8333333333

The i term is 2.833333333i.  Take half its coefficient (1.416666667).
Square it (2.006944445) and add it to both sides.

Add '2.006944445' to each side of the equation.
2.833333333i + 2.006944445 + i2 = 0.8333333333 + 2.006944445

Reorder the terms:
2.006944445 + 2.833333333i + i2 = 0.8333333333 + 2.006944445

Combine like terms: 0.8333333333 + 2.006944445 = 2.8402777783
2.006944445 + 2.833333333i + i2 = 2.8402777783

Factor a perfect square on the left side:
(i + 1.416666667)(i + 1.416666667) = 2.8402777783

Calculate the square root of the right side: 1.685312368

Break this problem into two subproblems by setting 
(i + 1.416666667) equal to 1.685312368 and -1.685312368.

Subproblem 1

i + 1.416666667 = 1.685312368 Simplifying i + 1.416666667 = 1.685312368 Reorder the terms: 1.416666667 + i = 1.685312368 Solving 1.416666667 + i = 1.685312368 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + i = 1.685312368 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + i = 1.685312368 + -1.416666667 i = 1.685312368 + -1.416666667 Combine like terms: 1.685312368 + -1.416666667 = 0.268645701 i = 0.268645701 Simplifying i = 0.268645701

Subproblem 2

i + 1.416666667 = -1.685312368 Simplifying i + 1.416666667 = -1.685312368 Reorder the terms: 1.416666667 + i = -1.685312368 Solving 1.416666667 + i = -1.685312368 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + i = -1.685312368 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + i = -1.685312368 + -1.416666667 i = -1.685312368 + -1.416666667 Combine like terms: -1.685312368 + -1.416666667 = -3.101979035 i = -3.101979035 Simplifying i = -3.101979035

Solution

The solution to the problem is based on the solutions from the subproblems. i = {0.268645701, -3.101979035}

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