13z^2+49z-2=0

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


Simplifying
13z2 + 49z + -2 = 0

Reorder the terms:
-2 + 49z + 13z2 = 0

Solving
-2 + 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'.
-0.1538461538 + 3.769230769z + z2 = 0

Move the constant term to the right:

Add '0.1538461538' to each side of the equation.
-0.1538461538 + 3.769230769z + 0.1538461538 + z2 = 0 + 0.1538461538

Reorder the terms:
-0.1538461538 + 0.1538461538 + 3.769230769z + z2 = 0 + 0.1538461538

Combine like terms: -0.1538461538 + 0.1538461538 = 0.0000000000
0.0000000000 + 3.769230769z + z2 = 0 + 0.1538461538
3.769230769z + z2 = 0 + 0.1538461538

Combine like terms: 0 + 0.1538461538 = 0.1538461538
3.769230769z + z2 = 0.1538461538

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 = 0.1538461538 + 3.551775149

Reorder the terms:
3.551775149 + 3.769230769z + z2 = 0.1538461538 + 3.551775149

Combine like terms: 0.1538461538 + 3.551775149 = 3.7056213028
3.551775149 + 3.769230769z + z2 = 3.7056213028

Factor a perfect square on the left side:
(z + 1.884615385)(z + 1.884615385) = 3.7056213028

Calculate the square root of the right side: 1.92499904

Break this problem into two subproblems by setting 
(z + 1.884615385) equal to 1.92499904 and -1.92499904.

Subproblem 1

z + 1.884615385 = 1.92499904 Simplifying z + 1.884615385 = 1.92499904 Reorder the terms: 1.884615385 + z = 1.92499904 Solving 1.884615385 + z = 1.92499904 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 = 1.92499904 + -1.884615385 Combine like terms: 1.884615385 + -1.884615385 = 0.000000000 0.000000000 + z = 1.92499904 + -1.884615385 z = 1.92499904 + -1.884615385 Combine like terms: 1.92499904 + -1.884615385 = 0.040383655 z = 0.040383655 Simplifying z = 0.040383655

Subproblem 2

z + 1.884615385 = -1.92499904 Simplifying z + 1.884615385 = -1.92499904 Reorder the terms: 1.884615385 + z = -1.92499904 Solving 1.884615385 + z = -1.92499904 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 = -1.92499904 + -1.884615385 Combine like terms: 1.884615385 + -1.884615385 = 0.000000000 0.000000000 + z = -1.92499904 + -1.884615385 z = -1.92499904 + -1.884615385 Combine like terms: -1.92499904 + -1.884615385 = -3.809614425 z = -3.809614425 Simplifying z = -3.809614425

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.040383655, -3.809614425}

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