0.5(8x^2)+0.5(14x(x+0.5))=24

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Solution for 0.5(8x^2)+0.5(14x(x+0.5))=24 equation:


Simplifying
0.5(8x2) + 0.5(14x(x + 0.5)) = 24

Remove parenthesis around (8x2)
0.5 * 8x2 + 0.5(14x(x + 0.5)) = 24

Multiply 0.5 * 8
4x2 + 0.5(14x(x + 0.5)) = 24

Reorder the terms:
4x2 + 0.5(14x(0.5 + x)) = 24
4x2 + 0.5((0.5 * 14x + x * 14x)) = 24
4x2 + 0.5((7x + 14x2)) = 24
4x2 + (7x * 0.5 + 14x2 * 0.5) = 24
4x2 + (3.5x + 7x2) = 24

Reorder the terms:
3.5x + 4x2 + 7x2 = 24

Combine like terms: 4x2 + 7x2 = 11x2
3.5x + 11x2 = 24

Solving
3.5x + 11x2 = 24

Solving for variable 'x'.

Reorder the terms:
-24 + 3.5x + 11x2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + 3.5x + 11x2 = 0

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

Divide each side by '11'.
-2.181818182 + 0.3181818182x + x2 = 0

Move the constant term to the right:

Add '2.181818182' to each side of the equation.
-2.181818182 + 0.3181818182x + 2.181818182 + x2 = 0 + 2.181818182

Reorder the terms:
-2.181818182 + 2.181818182 + 0.3181818182x + x2 = 0 + 2.181818182

Combine like terms: -2.181818182 + 2.181818182 = 0.000000000
0.000000000 + 0.3181818182x + x2 = 0 + 2.181818182
0.3181818182x + x2 = 0 + 2.181818182

Combine like terms: 0 + 2.181818182 = 2.181818182
0.3181818182x + x2 = 2.181818182

The x term is 0.3181818182x.  Take half its coefficient (0.1590909091).
Square it (0.02530991736) and add it to both sides.

Add '0.02530991736' to each side of the equation.
0.3181818182x + 0.02530991736 + x2 = 2.181818182 + 0.02530991736

Reorder the terms:
0.02530991736 + 0.3181818182x + x2 = 2.181818182 + 0.02530991736

Combine like terms: 2.181818182 + 0.02530991736 = 2.20712809936
0.02530991736 + 0.3181818182x + x2 = 2.20712809936

Factor a perfect square on the left side:
(x + 0.1590909091)(x + 0.1590909091) = 2.20712809936

Calculate the square root of the right side: 1.485640636

Break this problem into two subproblems by setting 
(x + 0.1590909091) equal to 1.485640636 and -1.485640636.

Subproblem 1

x + 0.1590909091 = 1.485640636 Simplifying x + 0.1590909091 = 1.485640636 Reorder the terms: 0.1590909091 + x = 1.485640636 Solving 0.1590909091 + x = 1.485640636 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1590909091' to each side of the equation. 0.1590909091 + -0.1590909091 + x = 1.485640636 + -0.1590909091 Combine like terms: 0.1590909091 + -0.1590909091 = 0.0000000000 0.0000000000 + x = 1.485640636 + -0.1590909091 x = 1.485640636 + -0.1590909091 Combine like terms: 1.485640636 + -0.1590909091 = 1.3265497269 x = 1.3265497269 Simplifying x = 1.3265497269

Subproblem 2

x + 0.1590909091 = -1.485640636 Simplifying x + 0.1590909091 = -1.485640636 Reorder the terms: 0.1590909091 + x = -1.485640636 Solving 0.1590909091 + x = -1.485640636 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1590909091' to each side of the equation. 0.1590909091 + -0.1590909091 + x = -1.485640636 + -0.1590909091 Combine like terms: 0.1590909091 + -0.1590909091 = 0.0000000000 0.0000000000 + x = -1.485640636 + -0.1590909091 x = -1.485640636 + -0.1590909091 Combine like terms: -1.485640636 + -0.1590909091 = -1.6447315451 x = -1.6447315451 Simplifying x = -1.6447315451

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3265497269, -1.6447315451}

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