z^2-2z-13=0

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Solution for z^2-2z-13=0 equation:


Simplifying
z2 + -2z + -13 = 0

Reorder the terms:
-13 + -2z + z2 = 0

Solving
-13 + -2z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + -2z + 13 + z2 = 0 + 13

Reorder the terms:
-13 + 13 + -2z + z2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + -2z + z2 = 0 + 13
-2z + z2 = 0 + 13

Combine like terms: 0 + 13 = 13
-2z + z2 = 13

The z term is -2z.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2z + 1 + z2 = 13 + 1

Reorder the terms:
1 + -2z + z2 = 13 + 1

Combine like terms: 13 + 1 = 14
1 + -2z + z2 = 14

Factor a perfect square on the left side:
(z + -1)(z + -1) = 14

Calculate the square root of the right side: 3.741657387

Break this problem into two subproblems by setting 
(z + -1) equal to 3.741657387 and -3.741657387.

Subproblem 1

z + -1 = 3.741657387 Simplifying z + -1 = 3.741657387 Reorder the terms: -1 + z = 3.741657387 Solving -1 + z = 3.741657387 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + z = 3.741657387 + 1 Combine like terms: -1 + 1 = 0 0 + z = 3.741657387 + 1 z = 3.741657387 + 1 Combine like terms: 3.741657387 + 1 = 4.741657387 z = 4.741657387 Simplifying z = 4.741657387

Subproblem 2

z + -1 = -3.741657387 Simplifying z + -1 = -3.741657387 Reorder the terms: -1 + z = -3.741657387 Solving -1 + z = -3.741657387 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + z = -3.741657387 + 1 Combine like terms: -1 + 1 = 0 0 + z = -3.741657387 + 1 z = -3.741657387 + 1 Combine like terms: -3.741657387 + 1 = -2.741657387 z = -2.741657387 Simplifying z = -2.741657387

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.741657387, -2.741657387}

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