5z^2+4z-13=0

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Solution for 5z^2+4z-13=0 equation:


Simplifying
5z2 + 4z + -13 = 0

Reorder the terms:
-13 + 4z + 5z2 = 0

Solving
-13 + 4z + 5z2 = 0

Solving for variable 'z'.

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

Divide each side by '5'.
-2.6 + 0.8z + z2 = 0

Move the constant term to the right:

Add '2.6' to each side of the equation.
-2.6 + 0.8z + 2.6 + z2 = 0 + 2.6

Reorder the terms:
-2.6 + 2.6 + 0.8z + z2 = 0 + 2.6

Combine like terms: -2.6 + 2.6 = 0.0
0.0 + 0.8z + z2 = 0 + 2.6
0.8z + z2 = 0 + 2.6

Combine like terms: 0 + 2.6 = 2.6
0.8z + z2 = 2.6

The z term is 0.8z.  Take half its coefficient (0.4).
Square it (0.16) and add it to both sides.

Add '0.16' to each side of the equation.
0.8z + 0.16 + z2 = 2.6 + 0.16

Reorder the terms:
0.16 + 0.8z + z2 = 2.6 + 0.16

Combine like terms: 2.6 + 0.16 = 2.76
0.16 + 0.8z + z2 = 2.76

Factor a perfect square on the left side:
(z + 0.4)(z + 0.4) = 2.76

Calculate the square root of the right side: 1.661324773

Break this problem into two subproblems by setting 
(z + 0.4) equal to 1.661324773 and -1.661324773.

Subproblem 1

z + 0.4 = 1.661324773 Simplifying z + 0.4 = 1.661324773 Reorder the terms: 0.4 + z = 1.661324773 Solving 0.4 + z = 1.661324773 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + z = 1.661324773 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + z = 1.661324773 + -0.4 z = 1.661324773 + -0.4 Combine like terms: 1.661324773 + -0.4 = 1.261324773 z = 1.261324773 Simplifying z = 1.261324773

Subproblem 2

z + 0.4 = -1.661324773 Simplifying z + 0.4 = -1.661324773 Reorder the terms: 0.4 + z = -1.661324773 Solving 0.4 + z = -1.661324773 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + z = -1.661324773 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + z = -1.661324773 + -0.4 z = -1.661324773 + -0.4 Combine like terms: -1.661324773 + -0.4 = -2.061324773 z = -2.061324773 Simplifying z = -2.061324773

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.261324773, -2.061324773}

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