4z^2+53=32z

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Solution for 4z^2+53=32z equation:


Simplifying
4z2 + 53 = 32z

Reorder the terms:
53 + 4z2 = 32z

Solving
53 + 4z2 = 32z

Solving for variable 'z'.

Reorder the terms:
53 + -32z + 4z2 = 32z + -32z

Combine like terms: 32z + -32z = 0
53 + -32z + 4z2 = 0

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

Divide each side by '4'.
13.25 + -8z + z2 = 0

Move the constant term to the right:

Add '-13.25' to each side of the equation.
13.25 + -8z + -13.25 + z2 = 0 + -13.25

Reorder the terms:
13.25 + -13.25 + -8z + z2 = 0 + -13.25

Combine like terms: 13.25 + -13.25 = 0.00
0.00 + -8z + z2 = 0 + -13.25
-8z + z2 = 0 + -13.25

Combine like terms: 0 + -13.25 = -13.25
-8z + z2 = -13.25

The z term is -8z.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8z + 16 + z2 = -13.25 + 16

Reorder the terms:
16 + -8z + z2 = -13.25 + 16

Combine like terms: -13.25 + 16 = 2.75
16 + -8z + z2 = 2.75

Factor a perfect square on the left side:
(z + -4)(z + -4) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(z + -4) equal to 1.658312395 and -1.658312395.

Subproblem 1

z + -4 = 1.658312395 Simplifying z + -4 = 1.658312395 Reorder the terms: -4 + z = 1.658312395 Solving -4 + z = 1.658312395 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = 1.658312395 + 4 Combine like terms: -4 + 4 = 0 0 + z = 1.658312395 + 4 z = 1.658312395 + 4 Combine like terms: 1.658312395 + 4 = 5.658312395 z = 5.658312395 Simplifying z = 5.658312395

Subproblem 2

z + -4 = -1.658312395 Simplifying z + -4 = -1.658312395 Reorder the terms: -4 + z = -1.658312395 Solving -4 + z = -1.658312395 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = -1.658312395 + 4 Combine like terms: -4 + 4 = 0 0 + z = -1.658312395 + 4 z = -1.658312395 + 4 Combine like terms: -1.658312395 + 4 = 2.341687605 z = 2.341687605 Simplifying z = 2.341687605

Solution

The solution to the problem is based on the solutions from the subproblems. z = {5.658312395, 2.341687605}

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