4z^2+31=24z

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Solution for 4z^2+31=24z equation:


Simplifying
4z2 + 31 = 24z

Reorder the terms:
31 + 4z2 = 24z

Solving
31 + 4z2 = 24z

Solving for variable 'z'.

Reorder the terms:
31 + -24z + 4z2 = 24z + -24z

Combine like terms: 24z + -24z = 0
31 + -24z + 4z2 = 0

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

Divide each side by '4'.
7.75 + -6z + z2 = 0

Move the constant term to the right:

Add '-7.75' to each side of the equation.
7.75 + -6z + -7.75 + z2 = 0 + -7.75

Reorder the terms:
7.75 + -7.75 + -6z + z2 = 0 + -7.75

Combine like terms: 7.75 + -7.75 = 0.00
0.00 + -6z + z2 = 0 + -7.75
-6z + z2 = 0 + -7.75

Combine like terms: 0 + -7.75 = -7.75
-6z + z2 = -7.75

The z term is -6z.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6z + 9 + z2 = -7.75 + 9

Reorder the terms:
9 + -6z + z2 = -7.75 + 9

Combine like terms: -7.75 + 9 = 1.25
9 + -6z + z2 = 1.25

Factor a perfect square on the left side:
(z + -3)(z + -3) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(z + -3) equal to 1.118033989 and -1.118033989.

Subproblem 1

z + -3 = 1.118033989 Simplifying z + -3 = 1.118033989 Reorder the terms: -3 + z = 1.118033989 Solving -3 + z = 1.118033989 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 1.118033989 + 3 Combine like terms: -3 + 3 = 0 0 + z = 1.118033989 + 3 z = 1.118033989 + 3 Combine like terms: 1.118033989 + 3 = 4.118033989 z = 4.118033989 Simplifying z = 4.118033989

Subproblem 2

z + -3 = -1.118033989 Simplifying z + -3 = -1.118033989 Reorder the terms: -3 + z = -1.118033989 Solving -3 + z = -1.118033989 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -1.118033989 + 3 Combine like terms: -3 + 3 = 0 0 + z = -1.118033989 + 3 z = -1.118033989 + 3 Combine like terms: -1.118033989 + 3 = 1.881966011 z = 1.881966011 Simplifying z = 1.881966011

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.118033989, 1.881966011}

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