7z^2-18z=19

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Solution for 7z^2-18z=19 equation:


Simplifying
7z2 + -18z = 19

Reorder the terms:
-18z + 7z2 = 19

Solving
-18z + 7z2 = 19

Solving for variable 'z'.

Reorder the terms:
-19 + -18z + 7z2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + -18z + 7z2 = 0

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

Divide each side by '7'.
-2.714285714 + -2.571428571z + z2 = 0

Move the constant term to the right:

Add '2.714285714' to each side of the equation.
-2.714285714 + -2.571428571z + 2.714285714 + z2 = 0 + 2.714285714

Reorder the terms:
-2.714285714 + 2.714285714 + -2.571428571z + z2 = 0 + 2.714285714

Combine like terms: -2.714285714 + 2.714285714 = 0.000000000
0.000000000 + -2.571428571z + z2 = 0 + 2.714285714
-2.571428571z + z2 = 0 + 2.714285714

Combine like terms: 0 + 2.714285714 = 2.714285714
-2.571428571z + z2 = 2.714285714

The z term is -2.571428571z.  Take half its coefficient (-1.285714286).
Square it (1.653061225) and add it to both sides.

Add '1.653061225' to each side of the equation.
-2.571428571z + 1.653061225 + z2 = 2.714285714 + 1.653061225

Reorder the terms:
1.653061225 + -2.571428571z + z2 = 2.714285714 + 1.653061225

Combine like terms: 2.714285714 + 1.653061225 = 4.367346939
1.653061225 + -2.571428571z + z2 = 4.367346939

Factor a perfect square on the left side:
(z + -1.285714286)(z + -1.285714286) = 4.367346939

Calculate the square root of the right side: 2.089819834

Break this problem into two subproblems by setting 
(z + -1.285714286) equal to 2.089819834 and -2.089819834.

Subproblem 1

z + -1.285714286 = 2.089819834 Simplifying z + -1.285714286 = 2.089819834 Reorder the terms: -1.285714286 + z = 2.089819834 Solving -1.285714286 + z = 2.089819834 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.285714286' to each side of the equation. -1.285714286 + 1.285714286 + z = 2.089819834 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + z = 2.089819834 + 1.285714286 z = 2.089819834 + 1.285714286 Combine like terms: 2.089819834 + 1.285714286 = 3.37553412 z = 3.37553412 Simplifying z = 3.37553412

Subproblem 2

z + -1.285714286 = -2.089819834 Simplifying z + -1.285714286 = -2.089819834 Reorder the terms: -1.285714286 + z = -2.089819834 Solving -1.285714286 + z = -2.089819834 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.285714286' to each side of the equation. -1.285714286 + 1.285714286 + z = -2.089819834 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + z = -2.089819834 + 1.285714286 z = -2.089819834 + 1.285714286 Combine like terms: -2.089819834 + 1.285714286 = -0.804105548 z = -0.804105548 Simplifying z = -0.804105548

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.37553412, -0.804105548}

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