4z^2+3z-5=0

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Solution for 4z^2+3z-5=0 equation:


Simplifying
4z2 + 3z + -5 = 0

Reorder the terms:
-5 + 3z + 4z2 = 0

Solving
-5 + 3z + 4z2 = 0

Solving for variable 'z'.

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

Divide each side by '4'.
-1.25 + 0.75z + z2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + 0.75z + 1.25 + z2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + 0.75z + z2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + 0.75z + z2 = 0 + 1.25
0.75z + z2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
0.75z + z2 = 1.25

The z term is 0.75z.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75z + 0.140625 + z2 = 1.25 + 0.140625

Reorder the terms:
0.140625 + 0.75z + z2 = 1.25 + 0.140625

Combine like terms: 1.25 + 0.140625 = 1.390625
0.140625 + 0.75z + z2 = 1.390625

Factor a perfect square on the left side:
(z + 0.375)(z + 0.375) = 1.390625

Calculate the square root of the right side: 1.179247642

Break this problem into two subproblems by setting 
(z + 0.375) equal to 1.179247642 and -1.179247642.

Subproblem 1

z + 0.375 = 1.179247642 Simplifying z + 0.375 = 1.179247642 Reorder the terms: 0.375 + z = 1.179247642 Solving 0.375 + z = 1.179247642 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + z = 1.179247642 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + z = 1.179247642 + -0.375 z = 1.179247642 + -0.375 Combine like terms: 1.179247642 + -0.375 = 0.804247642 z = 0.804247642 Simplifying z = 0.804247642

Subproblem 2

z + 0.375 = -1.179247642 Simplifying z + 0.375 = -1.179247642 Reorder the terms: 0.375 + z = -1.179247642 Solving 0.375 + z = -1.179247642 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + z = -1.179247642 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + z = -1.179247642 + -0.375 z = -1.179247642 + -0.375 Combine like terms: -1.179247642 + -0.375 = -1.554247642 z = -1.554247642 Simplifying z = -1.554247642

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.804247642, -1.554247642}

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