7+44z=32z^2

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Solution for 7+44z=32z^2 equation:


Simplifying
7 + 44z = 32z2

Solving
7 + 44z = 32z2

Solving for variable 'z'.

Combine like terms: 32z2 + -32z2 = 0
7 + 44z + -32z2 = 0

Begin completing the square.  Divide all terms by
-32 the coefficient of the squared term: 

Divide each side by '-32'.
-0.21875 + -1.375z + z2 = 0

Move the constant term to the right:

Add '0.21875' to each side of the equation.
-0.21875 + -1.375z + 0.21875 + z2 = 0 + 0.21875

Reorder the terms:
-0.21875 + 0.21875 + -1.375z + z2 = 0 + 0.21875

Combine like terms: -0.21875 + 0.21875 = 0.00000
0.00000 + -1.375z + z2 = 0 + 0.21875
-1.375z + z2 = 0 + 0.21875

Combine like terms: 0 + 0.21875 = 0.21875
-1.375z + z2 = 0.21875

The z term is -1.375z.  Take half its coefficient (-0.6875).
Square it (0.47265625) and add it to both sides.

Add '0.47265625' to each side of the equation.
-1.375z + 0.47265625 + z2 = 0.21875 + 0.47265625

Reorder the terms:
0.47265625 + -1.375z + z2 = 0.21875 + 0.47265625

Combine like terms: 0.21875 + 0.47265625 = 0.69140625
0.47265625 + -1.375z + z2 = 0.69140625

Factor a perfect square on the left side:
(z + -0.6875)(z + -0.6875) = 0.69140625

Calculate the square root of the right side: 0.831508418

Break this problem into two subproblems by setting 
(z + -0.6875) equal to 0.831508418 and -0.831508418.

Subproblem 1

z + -0.6875 = 0.831508418 Simplifying z + -0.6875 = 0.831508418 Reorder the terms: -0.6875 + z = 0.831508418 Solving -0.6875 + z = 0.831508418 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + z = 0.831508418 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + z = 0.831508418 + 0.6875 z = 0.831508418 + 0.6875 Combine like terms: 0.831508418 + 0.6875 = 1.519008418 z = 1.519008418 Simplifying z = 1.519008418

Subproblem 2

z + -0.6875 = -0.831508418 Simplifying z + -0.6875 = -0.831508418 Reorder the terms: -0.6875 + z = -0.831508418 Solving -0.6875 + z = -0.831508418 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + z = -0.831508418 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + z = -0.831508418 + 0.6875 z = -0.831508418 + 0.6875 Combine like terms: -0.831508418 + 0.6875 = -0.144008418 z = -0.144008418 Simplifying z = -0.144008418

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.519008418, -0.144008418}

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