13z^2+49z-8=0

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Solution for 13z^2+49z-8=0 equation:


Simplifying
13z2 + 49z + -8 = 0

Reorder the terms:
-8 + 49z + 13z2 = 0

Solving
-8 + 49z + 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'.
-0.6153846154 + 3.769230769z + z2 = 0

Move the constant term to the right:

Add '0.6153846154' to each side of the equation.
-0.6153846154 + 3.769230769z + 0.6153846154 + z2 = 0 + 0.6153846154

Reorder the terms:
-0.6153846154 + 0.6153846154 + 3.769230769z + z2 = 0 + 0.6153846154

Combine like terms: -0.6153846154 + 0.6153846154 = 0.0000000000
0.0000000000 + 3.769230769z + z2 = 0 + 0.6153846154
3.769230769z + z2 = 0 + 0.6153846154

Combine like terms: 0 + 0.6153846154 = 0.6153846154
3.769230769z + z2 = 0.6153846154

The z term is 3.769230769z.  Take half its coefficient (1.884615385).
Square it (3.551775149) and add it to both sides.

Add '3.551775149' to each side of the equation.
3.769230769z + 3.551775149 + z2 = 0.6153846154 + 3.551775149

Reorder the terms:
3.551775149 + 3.769230769z + z2 = 0.6153846154 + 3.551775149

Combine like terms: 0.6153846154 + 3.551775149 = 4.1671597644
3.551775149 + 3.769230769z + z2 = 4.1671597644

Factor a perfect square on the left side:
(z + 1.884615385)(z + 1.884615385) = 4.1671597644

Calculate the square root of the right side: 2.041362233

Break this problem into two subproblems by setting 
(z + 1.884615385) equal to 2.041362233 and -2.041362233.

Subproblem 1

z + 1.884615385 = 2.041362233 Simplifying z + 1.884615385 = 2.041362233 Reorder the terms: 1.884615385 + z = 2.041362233 Solving 1.884615385 + z = 2.041362233 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.884615385' to each side of the equation. 1.884615385 + -1.884615385 + z = 2.041362233 + -1.884615385 Combine like terms: 1.884615385 + -1.884615385 = 0.000000000 0.000000000 + z = 2.041362233 + -1.884615385 z = 2.041362233 + -1.884615385 Combine like terms: 2.041362233 + -1.884615385 = 0.156746848 z = 0.156746848 Simplifying z = 0.156746848

Subproblem 2

z + 1.884615385 = -2.041362233 Simplifying z + 1.884615385 = -2.041362233 Reorder the terms: 1.884615385 + z = -2.041362233 Solving 1.884615385 + z = -2.041362233 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.884615385' to each side of the equation. 1.884615385 + -1.884615385 + z = -2.041362233 + -1.884615385 Combine like terms: 1.884615385 + -1.884615385 = 0.000000000 0.000000000 + z = -2.041362233 + -1.884615385 z = -2.041362233 + -1.884615385 Combine like terms: -2.041362233 + -1.884615385 = -3.925977618 z = -3.925977618 Simplifying z = -3.925977618

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.156746848, -3.925977618}

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