7z^2-41z+48=0

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Solution for 7z^2-41z+48=0 equation:


Simplifying
7z2 + -41z + 48 = 0

Reorder the terms:
48 + -41z + 7z2 = 0

Solving
48 + -41z + 7z2 = 0

Solving for variable 'z'.

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

Divide each side by '7'.
6.857142857 + -5.857142857z + z2 = 0

Move the constant term to the right:

Add '-6.857142857' to each side of the equation.
6.857142857 + -5.857142857z + -6.857142857 + z2 = 0 + -6.857142857

Reorder the terms:
6.857142857 + -6.857142857 + -5.857142857z + z2 = 0 + -6.857142857

Combine like terms: 6.857142857 + -6.857142857 = 0.000000000
0.000000000 + -5.857142857z + z2 = 0 + -6.857142857
-5.857142857z + z2 = 0 + -6.857142857

Combine like terms: 0 + -6.857142857 = -6.857142857
-5.857142857z + z2 = -6.857142857

The z term is -5.857142857z.  Take half its coefficient (-2.928571429).
Square it (8.576530615) and add it to both sides.

Add '8.576530615' to each side of the equation.
-5.857142857z + 8.576530615 + z2 = -6.857142857 + 8.576530615

Reorder the terms:
8.576530615 + -5.857142857z + z2 = -6.857142857 + 8.576530615

Combine like terms: -6.857142857 + 8.576530615 = 1.719387758
8.576530615 + -5.857142857z + z2 = 1.719387758

Factor a perfect square on the left side:
(z + -2.928571429)(z + -2.928571429) = 1.719387758

Calculate the square root of the right side: 1.311254269

Break this problem into two subproblems by setting 
(z + -2.928571429) equal to 1.311254269 and -1.311254269.

Subproblem 1

z + -2.928571429 = 1.311254269 Simplifying z + -2.928571429 = 1.311254269 Reorder the terms: -2.928571429 + z = 1.311254269 Solving -2.928571429 + z = 1.311254269 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.928571429' to each side of the equation. -2.928571429 + 2.928571429 + z = 1.311254269 + 2.928571429 Combine like terms: -2.928571429 + 2.928571429 = 0.000000000 0.000000000 + z = 1.311254269 + 2.928571429 z = 1.311254269 + 2.928571429 Combine like terms: 1.311254269 + 2.928571429 = 4.239825698 z = 4.239825698 Simplifying z = 4.239825698

Subproblem 2

z + -2.928571429 = -1.311254269 Simplifying z + -2.928571429 = -1.311254269 Reorder the terms: -2.928571429 + z = -1.311254269 Solving -2.928571429 + z = -1.311254269 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.928571429' to each side of the equation. -2.928571429 + 2.928571429 + z = -1.311254269 + 2.928571429 Combine like terms: -2.928571429 + 2.928571429 = 0.000000000 0.000000000 + z = -1.311254269 + 2.928571429 z = -1.311254269 + 2.928571429 Combine like terms: -1.311254269 + 2.928571429 = 1.61731716 z = 1.61731716 Simplifying z = 1.61731716

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.239825698, 1.61731716}

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