14z^2-3=21z-7

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Solution for 14z^2-3=21z-7 equation:


Simplifying
14z2 + -3 = 21z + -7

Reorder the terms:
-3 + 14z2 = 21z + -7

Reorder the terms:
-3 + 14z2 = -7 + 21z

Solving
-3 + 14z2 = -7 + 21z

Solving for variable 'z'.

Reorder the terms:
-3 + 7 + -21z + 14z2 = -7 + 21z + 7 + -21z

Combine like terms: -3 + 7 = 4
4 + -21z + 14z2 = -7 + 21z + 7 + -21z

Reorder the terms:
4 + -21z + 14z2 = -7 + 7 + 21z + -21z

Combine like terms: -7 + 7 = 0
4 + -21z + 14z2 = 0 + 21z + -21z
4 + -21z + 14z2 = 21z + -21z

Combine like terms: 21z + -21z = 0
4 + -21z + 14z2 = 0

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

Divide each side by '14'.
0.2857142857 + -1.5z + z2 = 0

Move the constant term to the right:

Add '-0.2857142857' to each side of the equation.
0.2857142857 + -1.5z + -0.2857142857 + z2 = 0 + -0.2857142857

Reorder the terms:
0.2857142857 + -0.2857142857 + -1.5z + z2 = 0 + -0.2857142857

Combine like terms: 0.2857142857 + -0.2857142857 = 0.0000000000
0.0000000000 + -1.5z + z2 = 0 + -0.2857142857
-1.5z + z2 = 0 + -0.2857142857

Combine like terms: 0 + -0.2857142857 = -0.2857142857
-1.5z + z2 = -0.2857142857

The z term is -1.5z.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5z + 0.5625 + z2 = -0.2857142857 + 0.5625

Reorder the terms:
0.5625 + -1.5z + z2 = -0.2857142857 + 0.5625

Combine like terms: -0.2857142857 + 0.5625 = 0.2767857143
0.5625 + -1.5z + z2 = 0.2767857143

Factor a perfect square on the left side:
(z + -0.75)(z + -0.75) = 0.2767857143

Calculate the square root of the right side: 0.526104281

Break this problem into two subproblems by setting 
(z + -0.75) equal to 0.526104281 and -0.526104281.

Subproblem 1

z + -0.75 = 0.526104281 Simplifying z + -0.75 = 0.526104281 Reorder the terms: -0.75 + z = 0.526104281 Solving -0.75 + z = 0.526104281 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = 0.526104281 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 0.526104281 + 0.75 z = 0.526104281 + 0.75 Combine like terms: 0.526104281 + 0.75 = 1.276104281 z = 1.276104281 Simplifying z = 1.276104281

Subproblem 2

z + -0.75 = -0.526104281 Simplifying z + -0.75 = -0.526104281 Reorder the terms: -0.75 + z = -0.526104281 Solving -0.75 + z = -0.526104281 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = -0.526104281 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -0.526104281 + 0.75 z = -0.526104281 + 0.75 Combine like terms: -0.526104281 + 0.75 = 0.223895719 z = 0.223895719 Simplifying z = 0.223895719

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.276104281, 0.223895719}

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