5(4z+6)-2(z-4)=7z(z+4)-2(7z-2)-48

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Solution for 5(4z+6)-2(z-4)=7z(z+4)-2(7z-2)-48 equation:


Simplifying
5(4z + 6) + -2(z + -4) = 7z(z + 4) + -2(7z + -2) + -48

Reorder the terms:
5(6 + 4z) + -2(z + -4) = 7z(z + 4) + -2(7z + -2) + -48
(6 * 5 + 4z * 5) + -2(z + -4) = 7z(z + 4) + -2(7z + -2) + -48
(30 + 20z) + -2(z + -4) = 7z(z + 4) + -2(7z + -2) + -48

Reorder the terms:
30 + 20z + -2(-4 + z) = 7z(z + 4) + -2(7z + -2) + -48
30 + 20z + (-4 * -2 + z * -2) = 7z(z + 4) + -2(7z + -2) + -48
30 + 20z + (8 + -2z) = 7z(z + 4) + -2(7z + -2) + -48

Reorder the terms:
30 + 8 + 20z + -2z = 7z(z + 4) + -2(7z + -2) + -48

Combine like terms: 30 + 8 = 38
38 + 20z + -2z = 7z(z + 4) + -2(7z + -2) + -48

Combine like terms: 20z + -2z = 18z
38 + 18z = 7z(z + 4) + -2(7z + -2) + -48

Reorder the terms:
38 + 18z = 7z(4 + z) + -2(7z + -2) + -48
38 + 18z = (4 * 7z + z * 7z) + -2(7z + -2) + -48
38 + 18z = (28z + 7z2) + -2(7z + -2) + -48

Reorder the terms:
38 + 18z = 28z + 7z2 + -2(-2 + 7z) + -48
38 + 18z = 28z + 7z2 + (-2 * -2 + 7z * -2) + -48
38 + 18z = 28z + 7z2 + (4 + -14z) + -48

Reorder the terms:
38 + 18z = 4 + -48 + 28z + -14z + 7z2

Combine like terms: 4 + -48 = -44
38 + 18z = -44 + 28z + -14z + 7z2

Combine like terms: 28z + -14z = 14z
38 + 18z = -44 + 14z + 7z2

Solving
38 + 18z = -44 + 14z + 7z2

Solving for variable 'z'.

Reorder the terms:
38 + 44 + 18z + -14z + -7z2 = -44 + 14z + 7z2 + 44 + -14z + -7z2

Combine like terms: 38 + 44 = 82
82 + 18z + -14z + -7z2 = -44 + 14z + 7z2 + 44 + -14z + -7z2

Combine like terms: 18z + -14z = 4z
82 + 4z + -7z2 = -44 + 14z + 7z2 + 44 + -14z + -7z2

Reorder the terms:
82 + 4z + -7z2 = -44 + 44 + 14z + -14z + 7z2 + -7z2

Combine like terms: -44 + 44 = 0
82 + 4z + -7z2 = 0 + 14z + -14z + 7z2 + -7z2
82 + 4z + -7z2 = 14z + -14z + 7z2 + -7z2

Combine like terms: 14z + -14z = 0
82 + 4z + -7z2 = 0 + 7z2 + -7z2
82 + 4z + -7z2 = 7z2 + -7z2

Combine like terms: 7z2 + -7z2 = 0
82 + 4z + -7z2 = 0

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

Divide each side by '-7'.
-11.71428571 + -0.5714285714z + z2 = 0

Move the constant term to the right:

Add '11.71428571' to each side of the equation.
-11.71428571 + -0.5714285714z + 11.71428571 + z2 = 0 + 11.71428571

Reorder the terms:
-11.71428571 + 11.71428571 + -0.5714285714z + z2 = 0 + 11.71428571

Combine like terms: -11.71428571 + 11.71428571 = 0.00000000
0.00000000 + -0.5714285714z + z2 = 0 + 11.71428571
-0.5714285714z + z2 = 0 + 11.71428571

Combine like terms: 0 + 11.71428571 = 11.71428571
-0.5714285714z + z2 = 11.71428571

The z term is -0.5714285714z.  Take half its coefficient (-0.2857142857).
Square it (0.08163265305) and add it to both sides.

Add '0.08163265305' to each side of the equation.
-0.5714285714z + 0.08163265305 + z2 = 11.71428571 + 0.08163265305

Reorder the terms:
0.08163265305 + -0.5714285714z + z2 = 11.71428571 + 0.08163265305

Combine like terms: 11.71428571 + 0.08163265305 = 11.79591836305
0.08163265305 + -0.5714285714z + z2 = 11.79591836305

Factor a perfect square on the left side:
(z + -0.2857142857)(z + -0.2857142857) = 11.79591836305

Calculate the square root of the right side: 3.434518651

Break this problem into two subproblems by setting 
(z + -0.2857142857) equal to 3.434518651 and -3.434518651.

Subproblem 1

z + -0.2857142857 = 3.434518651 Simplifying z + -0.2857142857 = 3.434518651 Reorder the terms: -0.2857142857 + z = 3.434518651 Solving -0.2857142857 + z = 3.434518651 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + z = 3.434518651 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + z = 3.434518651 + 0.2857142857 z = 3.434518651 + 0.2857142857 Combine like terms: 3.434518651 + 0.2857142857 = 3.7202329367 z = 3.7202329367 Simplifying z = 3.7202329367

Subproblem 2

z + -0.2857142857 = -3.434518651 Simplifying z + -0.2857142857 = -3.434518651 Reorder the terms: -0.2857142857 + z = -3.434518651 Solving -0.2857142857 + z = -3.434518651 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + z = -3.434518651 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + z = -3.434518651 + 0.2857142857 z = -3.434518651 + 0.2857142857 Combine like terms: -3.434518651 + 0.2857142857 = -3.1488043653 z = -3.1488043653 Simplifying z = -3.1488043653

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.7202329367, -3.1488043653}

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