16z^2+59=64z

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Solution for 16z^2+59=64z equation:


Simplifying
16z2 + 59 = 64z

Reorder the terms:
59 + 16z2 = 64z

Solving
59 + 16z2 = 64z

Solving for variable 'z'.

Reorder the terms:
59 + -64z + 16z2 = 64z + -64z

Combine like terms: 64z + -64z = 0
59 + -64z + 16z2 = 0

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

Divide each side by '16'.
3.6875 + -4z + z2 = 0

Move the constant term to the right:

Add '-3.6875' to each side of the equation.
3.6875 + -4z + -3.6875 + z2 = 0 + -3.6875

Reorder the terms:
3.6875 + -3.6875 + -4z + z2 = 0 + -3.6875

Combine like terms: 3.6875 + -3.6875 = 0.0000
0.0000 + -4z + z2 = 0 + -3.6875
-4z + z2 = 0 + -3.6875

Combine like terms: 0 + -3.6875 = -3.6875
-4z + z2 = -3.6875

The z term is -4z.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4z + 4 + z2 = -3.6875 + 4

Reorder the terms:
4 + -4z + z2 = -3.6875 + 4

Combine like terms: -3.6875 + 4 = 0.3125
4 + -4z + z2 = 0.3125

Factor a perfect square on the left side:
(z + -2)(z + -2) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(z + -2) equal to 0.559016994 and -0.559016994.

Subproblem 1

z + -2 = 0.559016994 Simplifying z + -2 = 0.559016994 Reorder the terms: -2 + z = 0.559016994 Solving -2 + z = 0.559016994 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = 0.559016994 + 2 Combine like terms: -2 + 2 = 0 0 + z = 0.559016994 + 2 z = 0.559016994 + 2 Combine like terms: 0.559016994 + 2 = 2.559016994 z = 2.559016994 Simplifying z = 2.559016994

Subproblem 2

z + -2 = -0.559016994 Simplifying z + -2 = -0.559016994 Reorder the terms: -2 + z = -0.559016994 Solving -2 + z = -0.559016994 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = -0.559016994 + 2 Combine like terms: -2 + 2 = 0 0 + z = -0.559016994 + 2 z = -0.559016994 + 2 Combine like terms: -0.559016994 + 2 = 1.440983006 z = 1.440983006 Simplifying z = 1.440983006

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.559016994, 1.440983006}

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