13z^2-16z-4=0

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Solution for 13z^2-16z-4=0 equation:


Simplifying
13z2 + -16z + -4 = 0

Reorder the terms:
-4 + -16z + 13z2 = 0

Solving
-4 + -16z + 13z2 = 0

Solving for variable 'z'.

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

Divide each side by '13'.
-0.3076923077 + -1.230769231z + z2 = 0

Move the constant term to the right:

Add '0.3076923077' to each side of the equation.
-0.3076923077 + -1.230769231z + 0.3076923077 + z2 = 0 + 0.3076923077

Reorder the terms:
-0.3076923077 + 0.3076923077 + -1.230769231z + z2 = 0 + 0.3076923077

Combine like terms: -0.3076923077 + 0.3076923077 = 0.0000000000
0.0000000000 + -1.230769231z + z2 = 0 + 0.3076923077
-1.230769231z + z2 = 0 + 0.3076923077

Combine like terms: 0 + 0.3076923077 = 0.3076923077
-1.230769231z + z2 = 0.3076923077

The z term is -1.230769231z.  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.
-1.230769231z + 0.3786982250 + z2 = 0.3076923077 + 0.3786982250

Reorder the terms:
0.3786982250 + -1.230769231z + z2 = 0.3076923077 + 0.3786982250

Combine like terms: 0.3076923077 + 0.3786982250 = 0.6863905327
0.3786982250 + -1.230769231z + z2 = 0.6863905327

Factor a perfect square on the left side:
(z + -0.6153846155)(z + -0.6153846155) = 0.6863905327

Calculate the square root of the right side: 0.828486893

Break this problem into two subproblems by setting 
(z + -0.6153846155) equal to 0.828486893 and -0.828486893.

Subproblem 1

z + -0.6153846155 = 0.828486893 Simplifying z + -0.6153846155 = 0.828486893 Reorder the terms: -0.6153846155 + z = 0.828486893 Solving -0.6153846155 + z = 0.828486893 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.6153846155' to each side of the equation. -0.6153846155 + 0.6153846155 + z = 0.828486893 + 0.6153846155 Combine like terms: -0.6153846155 + 0.6153846155 = 0.0000000000 0.0000000000 + z = 0.828486893 + 0.6153846155 z = 0.828486893 + 0.6153846155 Combine like terms: 0.828486893 + 0.6153846155 = 1.4438715085 z = 1.4438715085 Simplifying z = 1.4438715085

Subproblem 2

z + -0.6153846155 = -0.828486893 Simplifying z + -0.6153846155 = -0.828486893 Reorder the terms: -0.6153846155 + z = -0.828486893 Solving -0.6153846155 + z = -0.828486893 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.6153846155' to each side of the equation. -0.6153846155 + 0.6153846155 + z = -0.828486893 + 0.6153846155 Combine like terms: -0.6153846155 + 0.6153846155 = 0.0000000000 0.0000000000 + z = -0.828486893 + 0.6153846155 z = -0.828486893 + 0.6153846155 Combine like terms: -0.828486893 + 0.6153846155 = -0.2131022775 z = -0.2131022775 Simplifying z = -0.2131022775

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.4438715085, -0.2131022775}

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