z(2z+7)=12

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Solution for z(2z+7)=12 equation:


Simplifying
z(2z + 7) = 12

Reorder the terms:
z(7 + 2z) = 12
(7 * z + 2z * z) = 12
(7z + 2z2) = 12

Solving
7z + 2z2 = 12

Solving for variable 'z'.

Reorder the terms:
-12 + 7z + 2z2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 7z + 2z2 = 0

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

Divide each side by '2'.
-6 + 3.5z + z2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 3.5z + 6 + z2 = 0 + 6

Reorder the terms:
-6 + 6 + 3.5z + z2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 3.5z + z2 = 0 + 6
3.5z + z2 = 0 + 6

Combine like terms: 0 + 6 = 6
3.5z + z2 = 6

The z term is 3.5z.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5z + 3.0625 + z2 = 6 + 3.0625

Reorder the terms:
3.0625 + 3.5z + z2 = 6 + 3.0625

Combine like terms: 6 + 3.0625 = 9.0625
3.0625 + 3.5z + z2 = 9.0625

Factor a perfect square on the left side:
(z + 1.75)(z + 1.75) = 9.0625

Calculate the square root of the right side: 3.010398645

Break this problem into two subproblems by setting 
(z + 1.75) equal to 3.010398645 and -3.010398645.

Subproblem 1

z + 1.75 = 3.010398645 Simplifying z + 1.75 = 3.010398645 Reorder the terms: 1.75 + z = 3.010398645 Solving 1.75 + z = 3.010398645 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + z = 3.010398645 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + z = 3.010398645 + -1.75 z = 3.010398645 + -1.75 Combine like terms: 3.010398645 + -1.75 = 1.260398645 z = 1.260398645 Simplifying z = 1.260398645

Subproblem 2

z + 1.75 = -3.010398645 Simplifying z + 1.75 = -3.010398645 Reorder the terms: 1.75 + z = -3.010398645 Solving 1.75 + z = -3.010398645 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + z = -3.010398645 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + z = -3.010398645 + -1.75 z = -3.010398645 + -1.75 Combine like terms: -3.010398645 + -1.75 = -4.760398645 z = -4.760398645 Simplifying z = -4.760398645

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.260398645, -4.760398645}

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