-0.5x+2.42(-0.5x^2+3x)=7.65

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Solution for -0.5x+2.42(-0.5x^2+3x)=7.65 equation:


Simplifying
-0.5x + 2.42(-0.5x2 + 3x) = 7.65

Reorder the terms:
-0.5x + 2.42(3x + -0.5x2) = 7.65
-0.5x + (3x * 2.42 + -0.5x2 * 2.42) = 7.65
-0.5x + (7.26x + -1.21x2) = 7.65

Combine like terms: -0.5x + 7.26x = 6.76x
6.76x + -1.21x2 = 7.65

Solving
6.76x + -1.21x2 = 7.65

Solving for variable 'x'.

Reorder the terms:
-7.65 + 6.76x + -1.21x2 = 7.65 + -7.65

Combine like terms: 7.65 + -7.65 = 0.00
-7.65 + 6.76x + -1.21x2 = 0.00

Begin completing the square.  Divide all terms by
-1.21 the coefficient of the squared term: 

Divide each side by '-1.21'.
6.32231405 + -5.58677686x + x2 = 0

Move the constant term to the right:

Add '-6.32231405' to each side of the equation.
6.32231405 + -5.58677686x + -6.32231405 + x2 = 0 + -6.32231405

Reorder the terms:
6.32231405 + -6.32231405 + -5.58677686x + x2 = 0 + -6.32231405

Combine like terms: 6.32231405 + -6.32231405 = 0.00000000
0.00000000 + -5.58677686x + x2 = 0 + -6.32231405
-5.58677686x + x2 = 0 + -6.32231405

Combine like terms: 0 + -6.32231405 = -6.32231405
-5.58677686x + x2 = -6.32231405

The x term is -5.58677686x.  Take half its coefficient (-2.79338843).
Square it (7.803018921) and add it to both sides.

Add '7.803018921' to each side of the equation.
-5.58677686x + 7.803018921 + x2 = -6.32231405 + 7.803018921

Reorder the terms:
7.803018921 + -5.58677686x + x2 = -6.32231405 + 7.803018921

Combine like terms: -6.32231405 + 7.803018921 = 1.480704871
7.803018921 + -5.58677686x + x2 = 1.480704871

Factor a perfect square on the left side:
(x + -2.79338843)(x + -2.79338843) = 1.480704871

Calculate the square root of the right side: 1.216842172

Break this problem into two subproblems by setting 
(x + -2.79338843) equal to 1.216842172 and -1.216842172.

Subproblem 1

x + -2.79338843 = 1.216842172 Simplifying x + -2.79338843 = 1.216842172 Reorder the terms: -2.79338843 + x = 1.216842172 Solving -2.79338843 + x = 1.216842172 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.79338843' to each side of the equation. -2.79338843 + 2.79338843 + x = 1.216842172 + 2.79338843 Combine like terms: -2.79338843 + 2.79338843 = 0.00000000 0.00000000 + x = 1.216842172 + 2.79338843 x = 1.216842172 + 2.79338843 Combine like terms: 1.216842172 + 2.79338843 = 4.010230602 x = 4.010230602 Simplifying x = 4.010230602

Subproblem 2

x + -2.79338843 = -1.216842172 Simplifying x + -2.79338843 = -1.216842172 Reorder the terms: -2.79338843 + x = -1.216842172 Solving -2.79338843 + x = -1.216842172 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.79338843' to each side of the equation. -2.79338843 + 2.79338843 + x = -1.216842172 + 2.79338843 Combine like terms: -2.79338843 + 2.79338843 = 0.00000000 0.00000000 + x = -1.216842172 + 2.79338843 x = -1.216842172 + 2.79338843 Combine like terms: -1.216842172 + 2.79338843 = 1.576546258 x = 1.576546258 Simplifying x = 1.576546258

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.010230602, 1.576546258}

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