7z^2+23z+1=0

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Solution for 7z^2+23z+1=0 equation:


Simplifying
7z2 + 23z + 1 = 0

Reorder the terms:
1 + 23z + 7z2 = 0

Solving
1 + 23z + 7z2 = 0

Solving for variable 'z'.

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

Divide each side by '7'.
0.1428571429 + 3.285714286z + z2 = 0

Move the constant term to the right:

Add '-0.1428571429' to each side of the equation.
0.1428571429 + 3.285714286z + -0.1428571429 + z2 = 0 + -0.1428571429

Reorder the terms:
0.1428571429 + -0.1428571429 + 3.285714286z + z2 = 0 + -0.1428571429

Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000
0.0000000000 + 3.285714286z + z2 = 0 + -0.1428571429
3.285714286z + z2 = 0 + -0.1428571429

Combine like terms: 0 + -0.1428571429 = -0.1428571429
3.285714286z + z2 = -0.1428571429

The z term is 3.285714286z.  Take half its coefficient (1.642857143).
Square it (2.698979592) and add it to both sides.

Add '2.698979592' to each side of the equation.
3.285714286z + 2.698979592 + z2 = -0.1428571429 + 2.698979592

Reorder the terms:
2.698979592 + 3.285714286z + z2 = -0.1428571429 + 2.698979592

Combine like terms: -0.1428571429 + 2.698979592 = 2.5561224491
2.698979592 + 3.285714286z + z2 = 2.5561224491

Factor a perfect square on the left side:
(z + 1.642857143)(z + 1.642857143) = 2.5561224491

Calculate the square root of the right side: 1.598787806

Break this problem into two subproblems by setting 
(z + 1.642857143) equal to 1.598787806 and -1.598787806.

Subproblem 1

z + 1.642857143 = 1.598787806 Simplifying z + 1.642857143 = 1.598787806 Reorder the terms: 1.642857143 + z = 1.598787806 Solving 1.642857143 + z = 1.598787806 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.642857143' to each side of the equation. 1.642857143 + -1.642857143 + z = 1.598787806 + -1.642857143 Combine like terms: 1.642857143 + -1.642857143 = 0.000000000 0.000000000 + z = 1.598787806 + -1.642857143 z = 1.598787806 + -1.642857143 Combine like terms: 1.598787806 + -1.642857143 = -0.044069337 z = -0.044069337 Simplifying z = -0.044069337

Subproblem 2

z + 1.642857143 = -1.598787806 Simplifying z + 1.642857143 = -1.598787806 Reorder the terms: 1.642857143 + z = -1.598787806 Solving 1.642857143 + z = -1.598787806 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.642857143' to each side of the equation. 1.642857143 + -1.642857143 + z = -1.598787806 + -1.642857143 Combine like terms: 1.642857143 + -1.642857143 = 0.000000000 0.000000000 + z = -1.598787806 + -1.642857143 z = -1.598787806 + -1.642857143 Combine like terms: -1.598787806 + -1.642857143 = -3.241644949 z = -3.241644949 Simplifying z = -3.241644949

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.044069337, -3.241644949}

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