2z^2=3z+15

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Solution for 2z^2=3z+15 equation:


Simplifying
2z2 = 3z + 15

Reorder the terms:
2z2 = 15 + 3z

Solving
2z2 = 15 + 3z

Solving for variable 'z'.

Reorder the terms:
-15 + -3z + 2z2 = 15 + 3z + -15 + -3z

Reorder the terms:
-15 + -3z + 2z2 = 15 + -15 + 3z + -3z

Combine like terms: 15 + -15 = 0
-15 + -3z + 2z2 = 0 + 3z + -3z
-15 + -3z + 2z2 = 3z + -3z

Combine like terms: 3z + -3z = 0
-15 + -3z + 2z2 = 0

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

Divide each side by '2'.
-7.5 + -1.5z + z2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + -1.5z + 7.5 + z2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + -1.5z + z2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + -1.5z + z2 = 0 + 7.5
-1.5z + z2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
-1.5z + z2 = 7.5

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 = 7.5 + 0.5625

Reorder the terms:
0.5625 + -1.5z + z2 = 7.5 + 0.5625

Combine like terms: 7.5 + 0.5625 = 8.0625
0.5625 + -1.5z + z2 = 8.0625

Factor a perfect square on the left side:
(z + -0.75)(z + -0.75) = 8.0625

Calculate the square root of the right side: 2.839454173

Break this problem into two subproblems by setting 
(z + -0.75) equal to 2.839454173 and -2.839454173.

Subproblem 1

z + -0.75 = 2.839454173 Simplifying z + -0.75 = 2.839454173 Reorder the terms: -0.75 + z = 2.839454173 Solving -0.75 + z = 2.839454173 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 = 2.839454173 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 2.839454173 + 0.75 z = 2.839454173 + 0.75 Combine like terms: 2.839454173 + 0.75 = 3.589454173 z = 3.589454173 Simplifying z = 3.589454173

Subproblem 2

z + -0.75 = -2.839454173 Simplifying z + -0.75 = -2.839454173 Reorder the terms: -0.75 + z = -2.839454173 Solving -0.75 + z = -2.839454173 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 = -2.839454173 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -2.839454173 + 0.75 z = -2.839454173 + 0.75 Combine like terms: -2.839454173 + 0.75 = -2.089454173 z = -2.089454173 Simplifying z = -2.089454173

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.589454173, -2.089454173}

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