(2+5i)(3-7i)=-1-i

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Solution for (2+5i)(3-7i)=-1-i equation:


Simplifying
(2 + 5i)(3 + -7i) = -1 + -1i

Multiply (2 + 5i) * (3 + -7i)
(2(3 + -7i) + 5i * (3 + -7i)) = -1 + -1i
((3 * 2 + -7i * 2) + 5i * (3 + -7i)) = -1 + -1i
((6 + -14i) + 5i * (3 + -7i)) = -1 + -1i
(6 + -14i + (3 * 5i + -7i * 5i)) = -1 + -1i
(6 + -14i + (15i + -35i2)) = -1 + -1i

Combine like terms: -14i + 15i = 1i
(6 + 1i + -35i2) = -1 + -1i

Solving
6 + 1i + -35i2 = -1 + -1i

Solving for variable 'i'.

Reorder the terms:
6 + 1 + 1i + i + -35i2 = -1 + -1i + 1 + i

Combine like terms: 6 + 1 = 7
7 + 1i + i + -35i2 = -1 + -1i + 1 + i

Combine like terms: 1i + i = 2i
7 + 2i + -35i2 = -1 + -1i + 1 + i

Reorder the terms:
7 + 2i + -35i2 = -1 + 1 + -1i + i

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

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

Begin completing the square.  Divide all terms by
-35 the coefficient of the squared term: 

Divide each side by '-35'.
-0.2 + -0.05714285714i + i2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -0.05714285714i + 0.2 + i2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -0.05714285714i + i2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -0.05714285714i + i2 = 0 + 0.2
-0.05714285714i + i2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-0.05714285714i + i2 = 0.2

The i term is -0.05714285714i.  Take half its coefficient (-0.02857142857).
Square it (0.0008163265305) and add it to both sides.

Add '0.0008163265305' to each side of the equation.
-0.05714285714i + 0.0008163265305 + i2 = 0.2 + 0.0008163265305

Reorder the terms:
0.0008163265305 + -0.05714285714i + i2 = 0.2 + 0.0008163265305

Combine like terms: 0.2 + 0.0008163265305 = 0.2008163265305
0.0008163265305 + -0.05714285714i + i2 = 0.2008163265305

Factor a perfect square on the left side:
(i + -0.02857142857)(i + -0.02857142857) = 0.2008163265305

Calculate the square root of the right side: 0.448125347

Break this problem into two subproblems by setting 
(i + -0.02857142857) equal to 0.448125347 and -0.448125347.

Subproblem 1

i + -0.02857142857 = 0.448125347 Simplifying i + -0.02857142857 = 0.448125347 Reorder the terms: -0.02857142857 + i = 0.448125347 Solving -0.02857142857 + i = 0.448125347 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.02857142857' to each side of the equation. -0.02857142857 + 0.02857142857 + i = 0.448125347 + 0.02857142857 Combine like terms: -0.02857142857 + 0.02857142857 = 0.00000000000 0.00000000000 + i = 0.448125347 + 0.02857142857 i = 0.448125347 + 0.02857142857 Combine like terms: 0.448125347 + 0.02857142857 = 0.47669677557 i = 0.47669677557 Simplifying i = 0.47669677557

Subproblem 2

i + -0.02857142857 = -0.448125347 Simplifying i + -0.02857142857 = -0.448125347 Reorder the terms: -0.02857142857 + i = -0.448125347 Solving -0.02857142857 + i = -0.448125347 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.02857142857' to each side of the equation. -0.02857142857 + 0.02857142857 + i = -0.448125347 + 0.02857142857 Combine like terms: -0.02857142857 + 0.02857142857 = 0.00000000000 0.00000000000 + i = -0.448125347 + 0.02857142857 i = -0.448125347 + 0.02857142857 Combine like terms: -0.448125347 + 0.02857142857 = -0.41955391843 i = -0.41955391843 Simplifying i = -0.41955391843

Solution

The solution to the problem is based on the solutions from the subproblems. i = {0.47669677557, -0.41955391843}

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