5x^2=5x+13

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Solution for 5x^2=5x+13 equation:


Simplifying
5x2 = 5x + 13

Reorder the terms:
5x2 = 13 + 5x

Solving
5x2 = 13 + 5x

Solving for variable 'x'.

Reorder the terms:
-13 + -5x + 5x2 = 13 + 5x + -13 + -5x

Reorder the terms:
-13 + -5x + 5x2 = 13 + -13 + 5x + -5x

Combine like terms: 13 + -13 = 0
-13 + -5x + 5x2 = 0 + 5x + -5x
-13 + -5x + 5x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
-13 + -5x + 5x2 = 0

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

Divide each side by '5'.
-2.6 + -1x + x2 = 0

Move the constant term to the right:

Add '2.6' to each side of the equation.
-2.6 + -1x + 2.6 + x2 = 0 + 2.6

Reorder the terms:
-2.6 + 2.6 + -1x + x2 = 0 + 2.6

Combine like terms: -2.6 + 2.6 = 0.0
0.0 + -1x + x2 = 0 + 2.6
-1x + x2 = 0 + 2.6

Combine like terms: 0 + 2.6 = 2.6
-1x + x2 = 2.6

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 2.6 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 2.6 + 0.25

Combine like terms: 2.6 + 0.25 = 2.85
0.25 + -1x + x2 = 2.85

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 2.85

Calculate the square root of the right side: 1.688194302

Break this problem into two subproblems by setting 
(x + -0.5) equal to 1.688194302 and -1.688194302.

Subproblem 1

x + -0.5 = 1.688194302 Simplifying x + -0.5 = 1.688194302 Reorder the terms: -0.5 + x = 1.688194302 Solving -0.5 + x = 1.688194302 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 1.688194302 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 1.688194302 + 0.5 x = 1.688194302 + 0.5 Combine like terms: 1.688194302 + 0.5 = 2.188194302 x = 2.188194302 Simplifying x = 2.188194302

Subproblem 2

x + -0.5 = -1.688194302 Simplifying x + -0.5 = -1.688194302 Reorder the terms: -0.5 + x = -1.688194302 Solving -0.5 + x = -1.688194302 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -1.688194302 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -1.688194302 + 0.5 x = -1.688194302 + 0.5 Combine like terms: -1.688194302 + 0.5 = -1.188194302 x = -1.188194302 Simplifying x = -1.188194302

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.188194302, -1.188194302}

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