z(2z+7)=7

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


Simplifying
z(2z + 7) = 7

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

Solving
7z + 2z2 = 7

Solving for variable 'z'.

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

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

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

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

Move the constant term to the right:

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

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

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

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

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 = 3.5 + 3.0625

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

Combine like terms: 3.5 + 3.0625 = 6.5625
3.0625 + 3.5z + z2 = 6.5625

Factor a perfect square on the left side:
(z + 1.75)(z + 1.75) = 6.5625

Calculate the square root of the right side: 2.561737691

Break this problem into two subproblems by setting 
(z + 1.75) equal to 2.561737691 and -2.561737691.

Subproblem 1

z + 1.75 = 2.561737691 Simplifying z + 1.75 = 2.561737691 Reorder the terms: 1.75 + z = 2.561737691 Solving 1.75 + z = 2.561737691 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 = 2.561737691 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + z = 2.561737691 + -1.75 z = 2.561737691 + -1.75 Combine like terms: 2.561737691 + -1.75 = 0.811737691 z = 0.811737691 Simplifying z = 0.811737691

Subproblem 2

z + 1.75 = -2.561737691 Simplifying z + 1.75 = -2.561737691 Reorder the terms: 1.75 + z = -2.561737691 Solving 1.75 + z = -2.561737691 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 = -2.561737691 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + z = -2.561737691 + -1.75 z = -2.561737691 + -1.75 Combine like terms: -2.561737691 + -1.75 = -4.311737691 z = -4.311737691 Simplifying z = -4.311737691

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.811737691, -4.311737691}

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