5z^2+8z-5=0

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Solution for 5z^2+8z-5=0 equation:


Simplifying
5z2 + 8z + -5 = 0

Reorder the terms:
-5 + 8z + 5z2 = 0

Solving
-5 + 8z + 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'.
-1 + 1.6z + z2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 1.6z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + 1.6z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 1.6z + z2 = 0 + 1
1.6z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
1.6z + z2 = 1

The z term is 1.6z.  Take half its coefficient (0.8).
Square it (0.64) and add it to both sides.

Add '0.64' to each side of the equation.
1.6z + 0.64 + z2 = 1 + 0.64

Reorder the terms:
0.64 + 1.6z + z2 = 1 + 0.64

Combine like terms: 1 + 0.64 = 1.64
0.64 + 1.6z + z2 = 1.64

Factor a perfect square on the left side:
(z + 0.8)(z + 0.8) = 1.64

Calculate the square root of the right side: 1.280624847

Break this problem into two subproblems by setting 
(z + 0.8) equal to 1.280624847 and -1.280624847.

Subproblem 1

z + 0.8 = 1.280624847 Simplifying z + 0.8 = 1.280624847 Reorder the terms: 0.8 + z = 1.280624847 Solving 0.8 + z = 1.280624847 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + z = 1.280624847 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + z = 1.280624847 + -0.8 z = 1.280624847 + -0.8 Combine like terms: 1.280624847 + -0.8 = 0.480624847 z = 0.480624847 Simplifying z = 0.480624847

Subproblem 2

z + 0.8 = -1.280624847 Simplifying z + 0.8 = -1.280624847 Reorder the terms: 0.8 + z = -1.280624847 Solving 0.8 + z = -1.280624847 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + z = -1.280624847 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + z = -1.280624847 + -0.8 z = -1.280624847 + -0.8 Combine like terms: -1.280624847 + -0.8 = -2.080624847 z = -2.080624847 Simplifying z = -2.080624847

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.480624847, -2.080624847}

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