-z^2+15z-13=0

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Solution for -z^2+15z-13=0 equation:


Simplifying
-1z2 + 15z + -13 = 0

Reorder the terms:
-13 + 15z + -1z2 = 0

Solving
-13 + 15z + -1z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
13 + -15z + z2 = 0

Move the constant term to the right:

Add '-13' to each side of the equation.
13 + -15z + -13 + z2 = 0 + -13

Reorder the terms:
13 + -13 + -15z + z2 = 0 + -13

Combine like terms: 13 + -13 = 0
0 + -15z + z2 = 0 + -13
-15z + z2 = 0 + -13

Combine like terms: 0 + -13 = -13
-15z + z2 = -13

The z term is -15z.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15z + 56.25 + z2 = -13 + 56.25

Reorder the terms:
56.25 + -15z + z2 = -13 + 56.25

Combine like terms: -13 + 56.25 = 43.25
56.25 + -15z + z2 = 43.25

Factor a perfect square on the left side:
(z + -7.5)(z + -7.5) = 43.25

Calculate the square root of the right side: 6.576473219

Break this problem into two subproblems by setting 
(z + -7.5) equal to 6.576473219 and -6.576473219.

Subproblem 1

z + -7.5 = 6.576473219 Simplifying z + -7.5 = 6.576473219 Reorder the terms: -7.5 + z = 6.576473219 Solving -7.5 + z = 6.576473219 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + z = 6.576473219 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + z = 6.576473219 + 7.5 z = 6.576473219 + 7.5 Combine like terms: 6.576473219 + 7.5 = 14.076473219 z = 14.076473219 Simplifying z = 14.076473219

Subproblem 2

z + -7.5 = -6.576473219 Simplifying z + -7.5 = -6.576473219 Reorder the terms: -7.5 + z = -6.576473219 Solving -7.5 + z = -6.576473219 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + z = -6.576473219 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + z = -6.576473219 + 7.5 z = -6.576473219 + 7.5 Combine like terms: -6.576473219 + 7.5 = 0.923526781 z = 0.923526781 Simplifying z = 0.923526781

Solution

The solution to the problem is based on the solutions from the subproblems. z = {14.076473219, 0.923526781}

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