0=4.5x^2-19x+1.9

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Solution for 0=4.5x^2-19x+1.9 equation:


Simplifying
0 = 4.5x2 + -19x + 1.9

Reorder the terms:
0 = 1.9 + -19x + 4.5x2

Solving
0 = 1.9 + -19x + 4.5x2

Solving for variable 'x'.

Combine like terms: 0 + -1.9 = -1.9
-1.9 + 19x + -4.5x2 = 1.9 + -19x + 4.5x2 + -1.9 + 19x + -4.5x2

Reorder the terms:
-1.9 + 19x + -4.5x2 = 1.9 + -1.9 + -19x + 19x + 4.5x2 + -4.5x2

Combine like terms: 1.9 + -1.9 = 0.0
-1.9 + 19x + -4.5x2 = 0.0 + -19x + 19x + 4.5x2 + -4.5x2
-1.9 + 19x + -4.5x2 = -19x + 19x + 4.5x2 + -4.5x2

Combine like terms: -19x + 19x = 0
-1.9 + 19x + -4.5x2 = 0 + 4.5x2 + -4.5x2
-1.9 + 19x + -4.5x2 = 4.5x2 + -4.5x2

Combine like terms: 4.5x2 + -4.5x2 = 0.0
-1.9 + 19x + -4.5x2 = 0.0

Begin completing the square.  Divide all terms by
-4.5 the coefficient of the squared term: 

Divide each side by '-4.5'.
0.4222222222 + -4.222222222x + x2 = 0

Move the constant term to the right:

Add '-0.4222222222' to each side of the equation.
0.4222222222 + -4.222222222x + -0.4222222222 + x2 = 0 + -0.4222222222

Reorder the terms:
0.4222222222 + -0.4222222222 + -4.222222222x + x2 = 0 + -0.4222222222

Combine like terms: 0.4222222222 + -0.4222222222 = 0.0000000000
0.0000000000 + -4.222222222x + x2 = 0 + -0.4222222222
-4.222222222x + x2 = 0 + -0.4222222222

Combine like terms: 0 + -0.4222222222 = -0.4222222222
-4.222222222x + x2 = -0.4222222222

The x term is -4.222222222x.  Take half its coefficient (-2.111111111).
Square it (4.456790123) and add it to both sides.

Add '4.456790123' to each side of the equation.
-4.222222222x + 4.456790123 + x2 = -0.4222222222 + 4.456790123

Reorder the terms:
4.456790123 + -4.222222222x + x2 = -0.4222222222 + 4.456790123

Combine like terms: -0.4222222222 + 4.456790123 = 4.0345679008
4.456790123 + -4.222222222x + x2 = 4.0345679008

Factor a perfect square on the left side:
(x + -2.111111111)(x + -2.111111111) = 4.0345679008

Calculate the square root of the right side: 2.008623385

Break this problem into two subproblems by setting 
(x + -2.111111111) equal to 2.008623385 and -2.008623385.

Subproblem 1

x + -2.111111111 = 2.008623385 Simplifying x + -2.111111111 = 2.008623385 Reorder the terms: -2.111111111 + x = 2.008623385 Solving -2.111111111 + x = 2.008623385 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.111111111' to each side of the equation. -2.111111111 + 2.111111111 + x = 2.008623385 + 2.111111111 Combine like terms: -2.111111111 + 2.111111111 = 0.000000000 0.000000000 + x = 2.008623385 + 2.111111111 x = 2.008623385 + 2.111111111 Combine like terms: 2.008623385 + 2.111111111 = 4.119734496 x = 4.119734496 Simplifying x = 4.119734496

Subproblem 2

x + -2.111111111 = -2.008623385 Simplifying x + -2.111111111 = -2.008623385 Reorder the terms: -2.111111111 + x = -2.008623385 Solving -2.111111111 + x = -2.008623385 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.111111111' to each side of the equation. -2.111111111 + 2.111111111 + x = -2.008623385 + 2.111111111 Combine like terms: -2.111111111 + 2.111111111 = 0.000000000 0.000000000 + x = -2.008623385 + 2.111111111 x = -2.008623385 + 2.111111111 Combine like terms: -2.008623385 + 2.111111111 = 0.102487726 x = 0.102487726 Simplifying x = 0.102487726

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.119734496, 0.102487726}

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