2z^2-5z-4=0

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


Simplifying
2z2 + -5z + -4 = 0

Reorder the terms:
-4 + -5z + 2z2 = 0

Solving
-4 + -5z + 2z2 = 0

Solving for variable 'z'.

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

Divide each side by '2'.
-2 + -2.5z + z2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -2.5z + 2 + z2 = 0 + 2

Reorder the terms:
-2 + 2 + -2.5z + z2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -2.5z + z2 = 0 + 2
-2.5z + z2 = 0 + 2

Combine like terms: 0 + 2 = 2
-2.5z + z2 = 2

The z term is -2.5z.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5z + 1.5625 + z2 = 2 + 1.5625

Reorder the terms:
1.5625 + -2.5z + z2 = 2 + 1.5625

Combine like terms: 2 + 1.5625 = 3.5625
1.5625 + -2.5z + z2 = 3.5625

Factor a perfect square on the left side:
(z + -1.25)(z + -1.25) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(z + -1.25) equal to 1.887458609 and -1.887458609.

Subproblem 1

z + -1.25 = 1.887458609 Simplifying z + -1.25 = 1.887458609 Reorder the terms: -1.25 + z = 1.887458609 Solving -1.25 + z = 1.887458609 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + z = 1.887458609 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + z = 1.887458609 + 1.25 z = 1.887458609 + 1.25 Combine like terms: 1.887458609 + 1.25 = 3.137458609 z = 3.137458609 Simplifying z = 3.137458609

Subproblem 2

z + -1.25 = -1.887458609 Simplifying z + -1.25 = -1.887458609 Reorder the terms: -1.25 + z = -1.887458609 Solving -1.25 + z = -1.887458609 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + z = -1.887458609 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + z = -1.887458609 + 1.25 z = -1.887458609 + 1.25 Combine like terms: -1.887458609 + 1.25 = -0.637458609 z = -0.637458609 Simplifying z = -0.637458609

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.137458609, -0.637458609}

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