2z+5=4z^2

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Solution for 2z+5=4z^2 equation:


Simplifying
2z + 5 = 4z2

Reorder the terms:
5 + 2z = 4z2

Solving
5 + 2z = 4z2

Solving for variable 'z'.

Combine like terms: 4z2 + -4z2 = 0
5 + 2z + -4z2 = 0

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

Divide each side by '-4'.
-1.25 + -0.5z + z2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + -0.5z + 1.25 + z2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + -0.5z + z2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + -0.5z + z2 = 0 + 1.25
-0.5z + z2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
-0.5z + z2 = 1.25

The z term is -0.5z.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5z + 0.0625 + z2 = 1.25 + 0.0625

Reorder the terms:
0.0625 + -0.5z + z2 = 1.25 + 0.0625

Combine like terms: 1.25 + 0.0625 = 1.3125
0.0625 + -0.5z + z2 = 1.3125

Factor a perfect square on the left side:
(z + -0.25)(z + -0.25) = 1.3125

Calculate the square root of the right side: 1.145643924

Break this problem into two subproblems by setting 
(z + -0.25) equal to 1.145643924 and -1.145643924.

Subproblem 1

z + -0.25 = 1.145643924 Simplifying z + -0.25 = 1.145643924 Reorder the terms: -0.25 + z = 1.145643924 Solving -0.25 + z = 1.145643924 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + z = 1.145643924 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = 1.145643924 + 0.25 z = 1.145643924 + 0.25 Combine like terms: 1.145643924 + 0.25 = 1.395643924 z = 1.395643924 Simplifying z = 1.395643924

Subproblem 2

z + -0.25 = -1.145643924 Simplifying z + -0.25 = -1.145643924 Reorder the terms: -0.25 + z = -1.145643924 Solving -0.25 + z = -1.145643924 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + z = -1.145643924 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + z = -1.145643924 + 0.25 z = -1.145643924 + 0.25 Combine like terms: -1.145643924 + 0.25 = -0.895643924 z = -0.895643924 Simplifying z = -0.895643924

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.395643924, -0.895643924}

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