2x^2+3x+1=1990

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Solution for 2x^2+3x+1=1990 equation:


Simplifying
2x2 + 3x + 1 = 1990

Reorder the terms:
1 + 3x + 2x2 = 1990

Solving
1 + 3x + 2x2 = 1990

Solving for variable 'x'.

Reorder the terms:
1 + -1990 + 3x + 2x2 = 1990 + -1990

Combine like terms: 1 + -1990 = -1989
-1989 + 3x + 2x2 = 1990 + -1990

Combine like terms: 1990 + -1990 = 0
-1989 + 3x + 2x2 = 0

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

Divide each side by '2'.
-994.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '994.5' to each side of the equation.
-994.5 + 1.5x + 994.5 + x2 = 0 + 994.5

Reorder the terms:
-994.5 + 994.5 + 1.5x + x2 = 0 + 994.5

Combine like terms: -994.5 + 994.5 = 0.0
0.0 + 1.5x + x2 = 0 + 994.5
1.5x + x2 = 0 + 994.5

Combine like terms: 0 + 994.5 = 994.5
1.5x + x2 = 994.5

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 994.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 994.5 + 0.5625

Combine like terms: 994.5 + 0.5625 = 995.0625
0.5625 + 1.5x + x2 = 995.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 995.0625

Calculate the square root of the right side: 31.544611267

Break this problem into two subproblems by setting 
(x + 0.75) equal to 31.544611267 and -31.544611267.

Subproblem 1

x + 0.75 = 31.544611267 Simplifying x + 0.75 = 31.544611267 Reorder the terms: 0.75 + x = 31.544611267 Solving 0.75 + x = 31.544611267 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 31.544611267 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 31.544611267 + -0.75 x = 31.544611267 + -0.75 Combine like terms: 31.544611267 + -0.75 = 30.794611267 x = 30.794611267 Simplifying x = 30.794611267

Subproblem 2

x + 0.75 = -31.544611267 Simplifying x + 0.75 = -31.544611267 Reorder the terms: 0.75 + x = -31.544611267 Solving 0.75 + x = -31.544611267 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -31.544611267 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -31.544611267 + -0.75 x = -31.544611267 + -0.75 Combine like terms: -31.544611267 + -0.75 = -32.294611267 x = -32.294611267 Simplifying x = -32.294611267

Solution

The solution to the problem is based on the solutions from the subproblems. x = {30.794611267, -32.294611267}

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