21z^2+8z-26=0

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Solution for 21z^2+8z-26=0 equation:


Simplifying
21z2 + 8z + -26 = 0

Reorder the terms:
-26 + 8z + 21z2 = 0

Solving
-26 + 8z + 21z2 = 0

Solving for variable 'z'.

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

Divide each side by '21'.
-1.238095238 + 0.380952381z + z2 = 0

Move the constant term to the right:

Add '1.238095238' to each side of the equation.
-1.238095238 + 0.380952381z + 1.238095238 + z2 = 0 + 1.238095238

Reorder the terms:
-1.238095238 + 1.238095238 + 0.380952381z + z2 = 0 + 1.238095238

Combine like terms: -1.238095238 + 1.238095238 = 0.000000000
0.000000000 + 0.380952381z + z2 = 0 + 1.238095238
0.380952381z + z2 = 0 + 1.238095238

Combine like terms: 0 + 1.238095238 = 1.238095238
0.380952381z + z2 = 1.238095238

The z term is 0.380952381z.  Take half its coefficient (0.1904761905).
Square it (0.03628117915) and add it to both sides.

Add '0.03628117915' to each side of the equation.
0.380952381z + 0.03628117915 + z2 = 1.238095238 + 0.03628117915

Reorder the terms:
0.03628117915 + 0.380952381z + z2 = 1.238095238 + 0.03628117915

Combine like terms: 1.238095238 + 0.03628117915 = 1.27437641715
0.03628117915 + 0.380952381z + z2 = 1.27437641715

Factor a perfect square on the left side:
(z + 0.1904761905)(z + 0.1904761905) = 1.27437641715

Calculate the square root of the right side: 1.128882818

Break this problem into two subproblems by setting 
(z + 0.1904761905) equal to 1.128882818 and -1.128882818.

Subproblem 1

z + 0.1904761905 = 1.128882818 Simplifying z + 0.1904761905 = 1.128882818 Reorder the terms: 0.1904761905 + z = 1.128882818 Solving 0.1904761905 + z = 1.128882818 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.1904761905' to each side of the equation. 0.1904761905 + -0.1904761905 + z = 1.128882818 + -0.1904761905 Combine like terms: 0.1904761905 + -0.1904761905 = 0.0000000000 0.0000000000 + z = 1.128882818 + -0.1904761905 z = 1.128882818 + -0.1904761905 Combine like terms: 1.128882818 + -0.1904761905 = 0.9384066275 z = 0.9384066275 Simplifying z = 0.9384066275

Subproblem 2

z + 0.1904761905 = -1.128882818 Simplifying z + 0.1904761905 = -1.128882818 Reorder the terms: 0.1904761905 + z = -1.128882818 Solving 0.1904761905 + z = -1.128882818 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.1904761905' to each side of the equation. 0.1904761905 + -0.1904761905 + z = -1.128882818 + -0.1904761905 Combine like terms: 0.1904761905 + -0.1904761905 = 0.0000000000 0.0000000000 + z = -1.128882818 + -0.1904761905 z = -1.128882818 + -0.1904761905 Combine like terms: -1.128882818 + -0.1904761905 = -1.3193590085 z = -1.3193590085 Simplifying z = -1.3193590085

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.9384066275, -1.3193590085}

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