4.5+2z=2z^2

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Solution for 4.5+2z=2z^2 equation:


Simplifying
4.5 + 2z = 2z2

Solving
4.5 + 2z = 2z2

Solving for variable 'z'.

Combine like terms: 2z2 + -2z2 = 0
4.5 + 2z + -2z2 = 0

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

Divide each side by '-2'.
-2.25 + -1z + z2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + -1z + 2.25 + z2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + -1z + z2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + -1z + z2 = 0 + 2.25
-1z + z2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
-1z + z2 = 2.25

The z term is -1z.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = 2.25 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 2.25 + 0.25

Combine like terms: 2.25 + 0.25 = 2.5
0.25 + -1z + z2 = 2.5

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 2.5

Calculate the square root of the right side: 1.58113883

Break this problem into two subproblems by setting 
(z + -0.5) equal to 1.58113883 and -1.58113883.

Subproblem 1

z + -0.5 = 1.58113883 Simplifying z + -0.5 = 1.58113883 Reorder the terms: -0.5 + z = 1.58113883 Solving -0.5 + z = 1.58113883 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = 1.58113883 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 1.58113883 + 0.5 z = 1.58113883 + 0.5 Combine like terms: 1.58113883 + 0.5 = 2.08113883 z = 2.08113883 Simplifying z = 2.08113883

Subproblem 2

z + -0.5 = -1.58113883 Simplifying z + -0.5 = -1.58113883 Reorder the terms: -0.5 + z = -1.58113883 Solving -0.5 + z = -1.58113883 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = -1.58113883 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -1.58113883 + 0.5 z = -1.58113883 + 0.5 Combine like terms: -1.58113883 + 0.5 = -1.08113883 z = -1.08113883 Simplifying z = -1.08113883

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.08113883, -1.08113883}

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