8z+25=4z^2

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Solution for 8z+25=4z^2 equation:


Simplifying
8z + 25 = 4z2

Reorder the terms:
25 + 8z = 4z2

Solving
25 + 8z = 4z2

Solving for variable 'z'.

Combine like terms: 4z2 + -4z2 = 0
25 + 8z + -4z2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-6.25 + -2z + z2 = 0

Move the constant term to the right:

Add '6.25' to each side of the equation.
-6.25 + -2z + 6.25 + z2 = 0 + 6.25

Reorder the terms:
-6.25 + 6.25 + -2z + z2 = 0 + 6.25

Combine like terms: -6.25 + 6.25 = 0.00
0.00 + -2z + z2 = 0 + 6.25
-2z + z2 = 0 + 6.25

Combine like terms: 0 + 6.25 = 6.25
-2z + z2 = 6.25

The z term is -2z.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2z + 1 + z2 = 6.25 + 1

Reorder the terms:
1 + -2z + z2 = 6.25 + 1

Combine like terms: 6.25 + 1 = 7.25
1 + -2z + z2 = 7.25

Factor a perfect square on the left side:
(z + -1)(z + -1) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(z + -1) equal to 2.692582404 and -2.692582404.

Subproblem 1

z + -1 = 2.692582404 Simplifying z + -1 = 2.692582404 Reorder the terms: -1 + z = 2.692582404 Solving -1 + z = 2.692582404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + z = 2.692582404 + 1 Combine like terms: -1 + 1 = 0 0 + z = 2.692582404 + 1 z = 2.692582404 + 1 Combine like terms: 2.692582404 + 1 = 3.692582404 z = 3.692582404 Simplifying z = 3.692582404

Subproblem 2

z + -1 = -2.692582404 Simplifying z + -1 = -2.692582404 Reorder the terms: -1 + z = -2.692582404 Solving -1 + z = -2.692582404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + z = -2.692582404 + 1 Combine like terms: -1 + 1 = 0 0 + z = -2.692582404 + 1 z = -2.692582404 + 1 Combine like terms: -2.692582404 + 1 = -1.692582404 z = -1.692582404 Simplifying z = -1.692582404

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.692582404, -1.692582404}

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