2z^2=3z+4

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Solution for 2z^2=3z+4 equation:


Simplifying
2z2 = 3z + 4

Reorder the terms:
2z2 = 4 + 3z

Solving
2z2 = 4 + 3z

Solving for variable 'z'.

Reorder the terms:
-4 + -3z + 2z2 = 4 + 3z + -4 + -3z

Reorder the terms:
-4 + -3z + 2z2 = 4 + -4 + 3z + -3z

Combine like terms: 4 + -4 = 0
-4 + -3z + 2z2 = 0 + 3z + -3z
-4 + -3z + 2z2 = 3z + -3z

Combine like terms: 3z + -3z = 0
-4 + -3z + 2z2 = 0

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

Divide each side by '2'.
-2 + -1.5z + z2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -1.5z + 2 + z2 = 0 + 2

Reorder the terms:
-2 + 2 + -1.5z + z2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1.5z + z2 = 0 + 2
-1.5z + z2 = 0 + 2

Combine like terms: 0 + 2 = 2
-1.5z + z2 = 2

The z term is -1.5z.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5z + 0.5625 + z2 = 2 + 0.5625

Reorder the terms:
0.5625 + -1.5z + z2 = 2 + 0.5625

Combine like terms: 2 + 0.5625 = 2.5625
0.5625 + -1.5z + z2 = 2.5625

Factor a perfect square on the left side:
(z + -0.75)(z + -0.75) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(z + -0.75) equal to 1.600781059 and -1.600781059.

Subproblem 1

z + -0.75 = 1.600781059 Simplifying z + -0.75 = 1.600781059 Reorder the terms: -0.75 + z = 1.600781059 Solving -0.75 + z = 1.600781059 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = 1.600781059 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 1.600781059 + 0.75 z = 1.600781059 + 0.75 Combine like terms: 1.600781059 + 0.75 = 2.350781059 z = 2.350781059 Simplifying z = 2.350781059

Subproblem 2

z + -0.75 = -1.600781059 Simplifying z + -0.75 = -1.600781059 Reorder the terms: -0.75 + z = -1.600781059 Solving -0.75 + z = -1.600781059 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = -1.600781059 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -1.600781059 + 0.75 z = -1.600781059 + 0.75 Combine like terms: -1.600781059 + 0.75 = -0.850781059 z = -0.850781059 Simplifying z = -0.850781059

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.350781059, -0.850781059}

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