13x^2=16x+5

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Solution for 13x^2=16x+5 equation:


Simplifying
13x2 = 16x + 5

Reorder the terms:
13x2 = 5 + 16x

Solving
13x2 = 5 + 16x

Solving for variable 'x'.

Reorder the terms:
-5 + -16x + 13x2 = 5 + 16x + -5 + -16x

Reorder the terms:
-5 + -16x + 13x2 = 5 + -5 + 16x + -16x

Combine like terms: 5 + -5 = 0
-5 + -16x + 13x2 = 0 + 16x + -16x
-5 + -16x + 13x2 = 16x + -16x

Combine like terms: 16x + -16x = 0
-5 + -16x + 13x2 = 0

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

Divide each side by '13'.
-0.3846153846 + -1.230769231x + x2 = 0

Move the constant term to the right:

Add '0.3846153846' to each side of the equation.
-0.3846153846 + -1.230769231x + 0.3846153846 + x2 = 0 + 0.3846153846

Reorder the terms:
-0.3846153846 + 0.3846153846 + -1.230769231x + x2 = 0 + 0.3846153846

Combine like terms: -0.3846153846 + 0.3846153846 = 0.0000000000
0.0000000000 + -1.230769231x + x2 = 0 + 0.3846153846
-1.230769231x + x2 = 0 + 0.3846153846

Combine like terms: 0 + 0.3846153846 = 0.3846153846
-1.230769231x + x2 = 0.3846153846

The x term is -1.230769231x.  Take half its coefficient (-0.6153846155).
Square it (0.3786982250) and add it to both sides.

Add '0.3786982250' to each side of the equation.
-1.230769231x + 0.3786982250 + x2 = 0.3846153846 + 0.3786982250

Reorder the terms:
0.3786982250 + -1.230769231x + x2 = 0.3846153846 + 0.3786982250

Combine like terms: 0.3846153846 + 0.3786982250 = 0.7633136096
0.3786982250 + -1.230769231x + x2 = 0.7633136096

Factor a perfect square on the left side:
(x + -0.6153846155)(x + -0.6153846155) = 0.7633136096

Calculate the square root of the right side: 0.873678207

Break this problem into two subproblems by setting 
(x + -0.6153846155) equal to 0.873678207 and -0.873678207.

Subproblem 1

x + -0.6153846155 = 0.873678207 Simplifying x + -0.6153846155 = 0.873678207 Reorder the terms: -0.6153846155 + x = 0.873678207 Solving -0.6153846155 + x = 0.873678207 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6153846155' to each side of the equation. -0.6153846155 + 0.6153846155 + x = 0.873678207 + 0.6153846155 Combine like terms: -0.6153846155 + 0.6153846155 = 0.0000000000 0.0000000000 + x = 0.873678207 + 0.6153846155 x = 0.873678207 + 0.6153846155 Combine like terms: 0.873678207 + 0.6153846155 = 1.4890628225 x = 1.4890628225 Simplifying x = 1.4890628225

Subproblem 2

x + -0.6153846155 = -0.873678207 Simplifying x + -0.6153846155 = -0.873678207 Reorder the terms: -0.6153846155 + x = -0.873678207 Solving -0.6153846155 + x = -0.873678207 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6153846155' to each side of the equation. -0.6153846155 + 0.6153846155 + x = -0.873678207 + 0.6153846155 Combine like terms: -0.6153846155 + 0.6153846155 = 0.0000000000 0.0000000000 + x = -0.873678207 + 0.6153846155 x = -0.873678207 + 0.6153846155 Combine like terms: -0.873678207 + 0.6153846155 = -0.2582935915 x = -0.2582935915 Simplifying x = -0.2582935915

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.4890628225, -0.2582935915}

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