23x+2x*x=67

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Solution for 23x+2x*x=67 equation:


Simplifying
23x + 2x * x = 67

Multiply x * x
23x + 2x2 = 67

Solving
23x + 2x2 = 67

Solving for variable 'x'.

Reorder the terms:
-67 + 23x + 2x2 = 67 + -67

Combine like terms: 67 + -67 = 0
-67 + 23x + 2x2 = 0

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

Divide each side by '2'.
-33.5 + 11.5x + x2 = 0

Move the constant term to the right:

Add '33.5' to each side of the equation.
-33.5 + 11.5x + 33.5 + x2 = 0 + 33.5

Reorder the terms:
-33.5 + 33.5 + 11.5x + x2 = 0 + 33.5

Combine like terms: -33.5 + 33.5 = 0.0
0.0 + 11.5x + x2 = 0 + 33.5
11.5x + x2 = 0 + 33.5

Combine like terms: 0 + 33.5 = 33.5
11.5x + x2 = 33.5

The x term is 11.5x.  Take half its coefficient (5.75).
Square it (33.0625) and add it to both sides.

Add '33.0625' to each side of the equation.
11.5x + 33.0625 + x2 = 33.5 + 33.0625

Reorder the terms:
33.0625 + 11.5x + x2 = 33.5 + 33.0625

Combine like terms: 33.5 + 33.0625 = 66.5625
33.0625 + 11.5x + x2 = 66.5625

Factor a perfect square on the left side:
(x + 5.75)(x + 5.75) = 66.5625

Calculate the square root of the right side: 8.158584436

Break this problem into two subproblems by setting 
(x + 5.75) equal to 8.158584436 and -8.158584436.

Subproblem 1

x + 5.75 = 8.158584436 Simplifying x + 5.75 = 8.158584436 Reorder the terms: 5.75 + x = 8.158584436 Solving 5.75 + x = 8.158584436 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + x = 8.158584436 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + x = 8.158584436 + -5.75 x = 8.158584436 + -5.75 Combine like terms: 8.158584436 + -5.75 = 2.408584436 x = 2.408584436 Simplifying x = 2.408584436

Subproblem 2

x + 5.75 = -8.158584436 Simplifying x + 5.75 = -8.158584436 Reorder the terms: 5.75 + x = -8.158584436 Solving 5.75 + x = -8.158584436 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + x = -8.158584436 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + x = -8.158584436 + -5.75 x = -8.158584436 + -5.75 Combine like terms: -8.158584436 + -5.75 = -13.908584436 x = -13.908584436 Simplifying x = -13.908584436

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.408584436, -13.908584436}

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