10i-8i^2=2.5

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Solution for 10i-8i^2=2.5 equation:


Simplifying
10i + -8i2 = 2.5

Solving
10i + -8i2 = 2.5

Solving for variable 'i'.

Reorder the terms:
-2.5 + 10i + -8i2 = 2.5 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
-2.5 + 10i + -8i2 = 0.0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
0.3125 + -1.25i + i2 = 0.0

Move the constant term to the right:

Add '-0.3125' to each side of the equation.
0.3125 + -1.25i + -0.3125 + i2 = 0.0 + -0.3125

Reorder the terms:
0.3125 + -0.3125 + -1.25i + i2 = 0.0 + -0.3125

Combine like terms: 0.3125 + -0.3125 = 0.0000
0.0000 + -1.25i + i2 = 0.0 + -0.3125
-1.25i + i2 = 0.0 + -0.3125

Combine like terms: 0.0 + -0.3125 = -0.3125
-1.25i + i2 = -0.3125

The i term is -1.25i.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25i + 0.390625 + i2 = -0.3125 + 0.390625

Reorder the terms:
0.390625 + -1.25i + i2 = -0.3125 + 0.390625

Combine like terms: -0.3125 + 0.390625 = 0.078125
0.390625 + -1.25i + i2 = 0.078125

Factor a perfect square on the left side:
(i + -0.625)(i + -0.625) = 0.078125

Calculate the square root of the right side: 0.279508497

Break this problem into two subproblems by setting 
(i + -0.625) equal to 0.279508497 and -0.279508497.

Subproblem 1

i + -0.625 = 0.279508497 Simplifying i + -0.625 = 0.279508497 Reorder the terms: -0.625 + i = 0.279508497 Solving -0.625 + i = 0.279508497 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + i = 0.279508497 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + i = 0.279508497 + 0.625 i = 0.279508497 + 0.625 Combine like terms: 0.279508497 + 0.625 = 0.904508497 i = 0.904508497 Simplifying i = 0.904508497

Subproblem 2

i + -0.625 = -0.279508497 Simplifying i + -0.625 = -0.279508497 Reorder the terms: -0.625 + i = -0.279508497 Solving -0.625 + i = -0.279508497 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + i = -0.279508497 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + i = -0.279508497 + 0.625 i = -0.279508497 + 0.625 Combine like terms: -0.279508497 + 0.625 = 0.345491503 i = 0.345491503 Simplifying i = 0.345491503

Solution

The solution to the problem is based on the solutions from the subproblems. i = {0.904508497, 0.345491503}

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