7=2z^2-3z

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Solution for 7=2z^2-3z equation:


Simplifying
7 = 2z2 + -3z

Reorder the terms:
7 = -3z + 2z2

Solving
7 = -3z + 2z2

Solving for variable 'z'.

Reorder the terms:
7 + 3z + -2z2 = -3z + 3z + 2z2 + -2z2

Combine like terms: -3z + 3z = 0
7 + 3z + -2z2 = 0 + 2z2 + -2z2
7 + 3z + -2z2 = 2z2 + -2z2

Combine like terms: 2z2 + -2z2 = 0
7 + 3z + -2z2 = 0

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

Divide each side by '-2'.
-3.5 + -1.5z + z2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + -1.5z + 3.5 + z2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + -1.5z + z2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + -1.5z + z2 = 0 + 3.5
-1.5z + z2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
-1.5z + z2 = 3.5

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 = 3.5 + 0.5625

Reorder the terms:
0.5625 + -1.5z + z2 = 3.5 + 0.5625

Combine like terms: 3.5 + 0.5625 = 4.0625
0.5625 + -1.5z + z2 = 4.0625

Factor a perfect square on the left side:
(z + -0.75)(z + -0.75) = 4.0625

Calculate the square root of the right side: 2.015564437

Break this problem into two subproblems by setting 
(z + -0.75) equal to 2.015564437 and -2.015564437.

Subproblem 1

z + -0.75 = 2.015564437 Simplifying z + -0.75 = 2.015564437 Reorder the terms: -0.75 + z = 2.015564437 Solving -0.75 + z = 2.015564437 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 = 2.015564437 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 2.015564437 + 0.75 z = 2.015564437 + 0.75 Combine like terms: 2.015564437 + 0.75 = 2.765564437 z = 2.765564437 Simplifying z = 2.765564437

Subproblem 2

z + -0.75 = -2.015564437 Simplifying z + -0.75 = -2.015564437 Reorder the terms: -0.75 + z = -2.015564437 Solving -0.75 + z = -2.015564437 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 = -2.015564437 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -2.015564437 + 0.75 z = -2.015564437 + 0.75 Combine like terms: -2.015564437 + 0.75 = -1.265564437 z = -1.265564437 Simplifying z = -1.265564437

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.765564437, -1.265564437}

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