7z^2-12z=20

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Solution for 7z^2-12z=20 equation:


Simplifying
7z2 + -12z = 20

Reorder the terms:
-12z + 7z2 = 20

Solving
-12z + 7z2 = 20

Solving for variable 'z'.

Reorder the terms:
-20 + -12z + 7z2 = 20 + -20

Combine like terms: 20 + -20 = 0
-20 + -12z + 7z2 = 0

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

Divide each side by '7'.
-2.857142857 + -1.714285714z + z2 = 0

Move the constant term to the right:

Add '2.857142857' to each side of the equation.
-2.857142857 + -1.714285714z + 2.857142857 + z2 = 0 + 2.857142857

Reorder the terms:
-2.857142857 + 2.857142857 + -1.714285714z + z2 = 0 + 2.857142857

Combine like terms: -2.857142857 + 2.857142857 = 0.000000000
0.000000000 + -1.714285714z + z2 = 0 + 2.857142857
-1.714285714z + z2 = 0 + 2.857142857

Combine like terms: 0 + 2.857142857 = 2.857142857
-1.714285714z + z2 = 2.857142857

The z term is -1.714285714z.  Take half its coefficient (-0.857142857).
Square it (0.7346938773) and add it to both sides.

Add '0.7346938773' to each side of the equation.
-1.714285714z + 0.7346938773 + z2 = 2.857142857 + 0.7346938773

Reorder the terms:
0.7346938773 + -1.714285714z + z2 = 2.857142857 + 0.7346938773

Combine like terms: 2.857142857 + 0.7346938773 = 3.5918367343
0.7346938773 + -1.714285714z + z2 = 3.5918367343

Factor a perfect square on the left side:
(z + -0.857142857)(z + -0.857142857) = 3.5918367343

Calculate the square root of the right side: 1.895214166

Break this problem into two subproblems by setting 
(z + -0.857142857) equal to 1.895214166 and -1.895214166.

Subproblem 1

z + -0.857142857 = 1.895214166 Simplifying z + -0.857142857 = 1.895214166 Reorder the terms: -0.857142857 + z = 1.895214166 Solving -0.857142857 + z = 1.895214166 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.857142857' to each side of the equation. -0.857142857 + 0.857142857 + z = 1.895214166 + 0.857142857 Combine like terms: -0.857142857 + 0.857142857 = 0.000000000 0.000000000 + z = 1.895214166 + 0.857142857 z = 1.895214166 + 0.857142857 Combine like terms: 1.895214166 + 0.857142857 = 2.752357023 z = 2.752357023 Simplifying z = 2.752357023

Subproblem 2

z + -0.857142857 = -1.895214166 Simplifying z + -0.857142857 = -1.895214166 Reorder the terms: -0.857142857 + z = -1.895214166 Solving -0.857142857 + z = -1.895214166 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.857142857' to each side of the equation. -0.857142857 + 0.857142857 + z = -1.895214166 + 0.857142857 Combine like terms: -0.857142857 + 0.857142857 = 0.000000000 0.000000000 + z = -1.895214166 + 0.857142857 z = -1.895214166 + 0.857142857 Combine like terms: -1.895214166 + 0.857142857 = -1.038071309 z = -1.038071309 Simplifying z = -1.038071309

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.752357023, -1.038071309}

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