26x(2-x)+(4x-5)=0

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Solution for 26x(2-x)+(4x-5)=0 equation:


Simplifying
26x(2 + -1x) + (4x + -5) = 0
(2 * 26x + -1x * 26x) + (4x + -5) = 0
(52x + -26x2) + (4x + -5) = 0

Reorder the terms:
52x + -26x2 + (-5 + 4x) = 0

Remove parenthesis around (-5 + 4x)
52x + -26x2 + -5 + 4x = 0

Reorder the terms:
-5 + 52x + 4x + -26x2 = 0

Combine like terms: 52x + 4x = 56x
-5 + 56x + -26x2 = 0

Solving
-5 + 56x + -26x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-26 the coefficient of the squared term: 

Divide each side by '-26'.
0.1923076923 + -2.153846154x + x2 = 0

Move the constant term to the right:

Add '-0.1923076923' to each side of the equation.
0.1923076923 + -2.153846154x + -0.1923076923 + x2 = 0 + -0.1923076923

Reorder the terms:
0.1923076923 + -0.1923076923 + -2.153846154x + x2 = 0 + -0.1923076923

Combine like terms: 0.1923076923 + -0.1923076923 = 0.0000000000
0.0000000000 + -2.153846154x + x2 = 0 + -0.1923076923
-2.153846154x + x2 = 0 + -0.1923076923

Combine like terms: 0 + -0.1923076923 = -0.1923076923
-2.153846154x + x2 = -0.1923076923

The x term is -2.153846154x.  Take half its coefficient (-1.076923077).
Square it (1.159763314) and add it to both sides.

Add '1.159763314' to each side of the equation.
-2.153846154x + 1.159763314 + x2 = -0.1923076923 + 1.159763314

Reorder the terms:
1.159763314 + -2.153846154x + x2 = -0.1923076923 + 1.159763314

Combine like terms: -0.1923076923 + 1.159763314 = 0.9674556217
1.159763314 + -2.153846154x + x2 = 0.9674556217

Factor a perfect square on the left side:
(x + -1.076923077)(x + -1.076923077) = 0.9674556217

Calculate the square root of the right side: 0.98359322

Break this problem into two subproblems by setting 
(x + -1.076923077) equal to 0.98359322 and -0.98359322.

Subproblem 1

x + -1.076923077 = 0.98359322 Simplifying x + -1.076923077 = 0.98359322 Reorder the terms: -1.076923077 + x = 0.98359322 Solving -1.076923077 + x = 0.98359322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.076923077' to each side of the equation. -1.076923077 + 1.076923077 + x = 0.98359322 + 1.076923077 Combine like terms: -1.076923077 + 1.076923077 = 0.000000000 0.000000000 + x = 0.98359322 + 1.076923077 x = 0.98359322 + 1.076923077 Combine like terms: 0.98359322 + 1.076923077 = 2.060516297 x = 2.060516297 Simplifying x = 2.060516297

Subproblem 2

x + -1.076923077 = -0.98359322 Simplifying x + -1.076923077 = -0.98359322 Reorder the terms: -1.076923077 + x = -0.98359322 Solving -1.076923077 + x = -0.98359322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.076923077' to each side of the equation. -1.076923077 + 1.076923077 + x = -0.98359322 + 1.076923077 Combine like terms: -1.076923077 + 1.076923077 = 0.000000000 0.000000000 + x = -0.98359322 + 1.076923077 x = -0.98359322 + 1.076923077 Combine like terms: -0.98359322 + 1.076923077 = 0.093329857 x = 0.093329857 Simplifying x = 0.093329857

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.060516297, 0.093329857}

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