23x+3x^2=1600

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Solution for 23x+3x^2=1600 equation: 


Simplifying
23x + 3x2 = 1600

Solving
23x + 3x2 = 1600

Solving for variable 'x'.

Reorder the terms:
-1600 + 23x + 3x2 = 1600 + -1600

Combine like terms: 1600 + -1600 = 0
-1600 + 23x + 3x2 = 0

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

Divide each side by '3'.
-533.3333333 + 7.666666667x + x2 = 0

Move the constant term to the right:

Add '533.3333333' to each side of the equation.
-533.3333333 + 7.666666667x + 533.3333333 + x2 = 0 + 533.3333333

Reorder the terms:
-533.3333333 + 533.3333333 + 7.666666667x + x2 = 0 + 533.3333333

Combine like terms: -533.3333333 + 533.3333333 = 0.0000000
0.0000000 + 7.666666667x + x2 = 0 + 533.3333333
7.666666667x + x2 = 0 + 533.3333333

Combine like terms: 0 + 533.3333333 = 533.3333333
7.666666667x + x2 = 533.3333333

The x term is 7.666666667x.  Take half its coefficient (3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
7.666666667x + 14.69444445 + x2 = 533.3333333 + 14.69444445

Reorder the terms:
14.69444445 + 7.666666667x + x2 = 533.3333333 + 14.69444445

Combine like terms: 533.3333333 + 14.69444445 = 548.02777775
14.69444445 + 7.666666667x + x2 = 548.02777775

Factor a perfect square on the left side:
(x + 3.833333334)(x + 3.833333334) = 548.02777775

Calculate the square root of the right side: 23.409993117

Break this problem into two subproblems by setting 
(x + 3.833333334) equal to 23.409993117 and -23.409993117.


Subproblem 1
x + 3.833333334 = 23.409993117

Simplifying
x + 3.833333334 = 23.409993117

Reorder the terms:
3.833333334 + x = 23.409993117

Solving
3.833333334 + x = 23.409993117

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3.833333334' to each side of the equation.
3.833333334 + -3.833333334 + x = 23.409993117 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + x = 23.409993117 + -3.833333334
x = 23.409993117 + -3.833333334

Combine like terms: 23.409993117 + -3.833333334 = 19.576659783
x = 19.576659783

Simplifying
x = 19.576659783


Subproblem 2
x + 3.833333334 = -23.409993117

Simplifying
x + 3.833333334 = -23.409993117

Reorder the terms:
3.833333334 + x = -23.409993117

Solving
3.833333334 + x = -23.409993117

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3.833333334' to each side of the equation.
3.833333334 + -3.833333334 + x = -23.409993117 + -3.833333334

Combine like terms: 3.833333334 + -3.833333334 = 0.000000000
0.000000000 + x = -23.409993117 + -3.833333334
x = -23.409993117 + -3.833333334

Combine like terms: -23.409993117 + -3.833333334 = -27.243326451
x = -27.243326451

Simplifying
x = -27.243326451


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {19.576659783, -27.243326451}

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