7z^2-11z-13=12-9z

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Solution for 7z^2-11z-13=12-9z equation:


Simplifying
7z2 + -11z + -13 = 12 + -9z

Reorder the terms:
-13 + -11z + 7z2 = 12 + -9z

Solving
-13 + -11z + 7z2 = 12 + -9z

Solving for variable 'z'.

Reorder the terms:
-13 + -12 + -11z + 9z + 7z2 = 12 + -9z + -12 + 9z

Combine like terms: -13 + -12 = -25
-25 + -11z + 9z + 7z2 = 12 + -9z + -12 + 9z

Combine like terms: -11z + 9z = -2z
-25 + -2z + 7z2 = 12 + -9z + -12 + 9z

Reorder the terms:
-25 + -2z + 7z2 = 12 + -12 + -9z + 9z

Combine like terms: 12 + -12 = 0
-25 + -2z + 7z2 = 0 + -9z + 9z
-25 + -2z + 7z2 = -9z + 9z

Combine like terms: -9z + 9z = 0
-25 + -2z + 7z2 = 0

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

Divide each side by '7'.
-3.571428571 + -0.2857142857z + z2 = 0

Move the constant term to the right:

Add '3.571428571' to each side of the equation.
-3.571428571 + -0.2857142857z + 3.571428571 + z2 = 0 + 3.571428571

Reorder the terms:
-3.571428571 + 3.571428571 + -0.2857142857z + z2 = 0 + 3.571428571

Combine like terms: -3.571428571 + 3.571428571 = 0.000000000
0.000000000 + -0.2857142857z + z2 = 0 + 3.571428571
-0.2857142857z + z2 = 0 + 3.571428571

Combine like terms: 0 + 3.571428571 = 3.571428571
-0.2857142857z + z2 = 3.571428571

The z term is -0.2857142857z.  Take half its coefficient (-0.1428571429).
Square it (0.02040816328) and add it to both sides.

Add '0.02040816328' to each side of the equation.
-0.2857142857z + 0.02040816328 + z2 = 3.571428571 + 0.02040816328

Reorder the terms:
0.02040816328 + -0.2857142857z + z2 = 3.571428571 + 0.02040816328

Combine like terms: 3.571428571 + 0.02040816328 = 3.59183673428
0.02040816328 + -0.2857142857z + z2 = 3.59183673428

Factor a perfect square on the left side:
(z + -0.1428571429)(z + -0.1428571429) = 3.59183673428

Calculate the square root of the right side: 1.895214166

Break this problem into two subproblems by setting 
(z + -0.1428571429) equal to 1.895214166 and -1.895214166.

Subproblem 1

z + -0.1428571429 = 1.895214166 Simplifying z + -0.1428571429 = 1.895214166 Reorder the terms: -0.1428571429 + z = 1.895214166 Solving -0.1428571429 + z = 1.895214166 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + z = 1.895214166 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + z = 1.895214166 + 0.1428571429 z = 1.895214166 + 0.1428571429 Combine like terms: 1.895214166 + 0.1428571429 = 2.0380713089 z = 2.0380713089 Simplifying z = 2.0380713089

Subproblem 2

z + -0.1428571429 = -1.895214166 Simplifying z + -0.1428571429 = -1.895214166 Reorder the terms: -0.1428571429 + z = -1.895214166 Solving -0.1428571429 + z = -1.895214166 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.1428571429' to each side of the equation. -0.1428571429 + 0.1428571429 + z = -1.895214166 + 0.1428571429 Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000 0.0000000000 + z = -1.895214166 + 0.1428571429 z = -1.895214166 + 0.1428571429 Combine like terms: -1.895214166 + 0.1428571429 = -1.7523570231 z = -1.7523570231 Simplifying z = -1.7523570231

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.0380713089, -1.7523570231}

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