z^2-14z+47=0

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Solution for z^2-14z+47=0 equation:


Simplifying
z2 + -14z + 47 = 0

Reorder the terms:
47 + -14z + z2 = 0

Solving
47 + -14z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-47' to each side of the equation.
47 + -14z + -47 + z2 = 0 + -47

Reorder the terms:
47 + -47 + -14z + z2 = 0 + -47

Combine like terms: 47 + -47 = 0
0 + -14z + z2 = 0 + -47
-14z + z2 = 0 + -47

Combine like terms: 0 + -47 = -47
-14z + z2 = -47

The z term is -14z.  Take half its coefficient (-7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
-14z + 49 + z2 = -47 + 49

Reorder the terms:
49 + -14z + z2 = -47 + 49

Combine like terms: -47 + 49 = 2
49 + -14z + z2 = 2

Factor a perfect square on the left side:
(z + -7)(z + -7) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(z + -7) equal to 1.414213562 and -1.414213562.

Subproblem 1

z + -7 = 1.414213562 Simplifying z + -7 = 1.414213562 Reorder the terms: -7 + z = 1.414213562 Solving -7 + z = 1.414213562 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + z = 1.414213562 + 7 Combine like terms: -7 + 7 = 0 0 + z = 1.414213562 + 7 z = 1.414213562 + 7 Combine like terms: 1.414213562 + 7 = 8.414213562 z = 8.414213562 Simplifying z = 8.414213562

Subproblem 2

z + -7 = -1.414213562 Simplifying z + -7 = -1.414213562 Reorder the terms: -7 + z = -1.414213562 Solving -7 + z = -1.414213562 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + z = -1.414213562 + 7 Combine like terms: -7 + 7 = 0 0 + z = -1.414213562 + 7 z = -1.414213562 + 7 Combine like terms: -1.414213562 + 7 = 5.585786438 z = 5.585786438 Simplifying z = 5.585786438

Solution

The solution to the problem is based on the solutions from the subproblems. z = {8.414213562, 5.585786438}

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