5z^2=7z+2

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Solution for 5z^2=7z+2 equation:


Simplifying
5z2 = 7z + 2

Reorder the terms:
5z2 = 2 + 7z

Solving
5z2 = 2 + 7z

Solving for variable 'z'.

Reorder the terms:
-2 + -7z + 5z2 = 2 + 7z + -2 + -7z

Reorder the terms:
-2 + -7z + 5z2 = 2 + -2 + 7z + -7z

Combine like terms: 2 + -2 = 0
-2 + -7z + 5z2 = 0 + 7z + -7z
-2 + -7z + 5z2 = 7z + -7z

Combine like terms: 7z + -7z = 0
-2 + -7z + 5z2 = 0

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

Divide each side by '5'.
-0.4 + -1.4z + z2 = 0

Move the constant term to the right:

Add '0.4' to each side of the equation.
-0.4 + -1.4z + 0.4 + z2 = 0 + 0.4

Reorder the terms:
-0.4 + 0.4 + -1.4z + z2 = 0 + 0.4

Combine like terms: -0.4 + 0.4 = 0.0
0.0 + -1.4z + z2 = 0 + 0.4
-1.4z + z2 = 0 + 0.4

Combine like terms: 0 + 0.4 = 0.4
-1.4z + z2 = 0.4

The z term is -1.4z.  Take half its coefficient (-0.7).
Square it (0.49) and add it to both sides.

Add '0.49' to each side of the equation.
-1.4z + 0.49 + z2 = 0.4 + 0.49

Reorder the terms:
0.49 + -1.4z + z2 = 0.4 + 0.49

Combine like terms: 0.4 + 0.49 = 0.89
0.49 + -1.4z + z2 = 0.89

Factor a perfect square on the left side:
(z + -0.7)(z + -0.7) = 0.89

Calculate the square root of the right side: 0.943398113

Break this problem into two subproblems by setting 
(z + -0.7) equal to 0.943398113 and -0.943398113.

Subproblem 1

z + -0.7 = 0.943398113 Simplifying z + -0.7 = 0.943398113 Reorder the terms: -0.7 + z = 0.943398113 Solving -0.7 + z = 0.943398113 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + z = 0.943398113 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + z = 0.943398113 + 0.7 z = 0.943398113 + 0.7 Combine like terms: 0.943398113 + 0.7 = 1.643398113 z = 1.643398113 Simplifying z = 1.643398113

Subproblem 2

z + -0.7 = -0.943398113 Simplifying z + -0.7 = -0.943398113 Reorder the terms: -0.7 + z = -0.943398113 Solving -0.7 + z = -0.943398113 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + z = -0.943398113 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + z = -0.943398113 + 0.7 z = -0.943398113 + 0.7 Combine like terms: -0.943398113 + 0.7 = -0.243398113 z = -0.243398113 Simplifying z = -0.243398113

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.643398113, -0.243398113}

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