2x^2+3x+1=4

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Solution for 2x^2+3x+1=4 equation:


Simplifying
2x2 + 3x + 1 = 4

Reorder the terms:
1 + 3x + 2x2 = 4

Solving
1 + 3x + 2x2 = 4

Solving for variable 'x'.

Reorder the terms:
1 + -4 + 3x + 2x2 = 4 + -4

Combine like terms: 1 + -4 = -3
-3 + 3x + 2x2 = 4 + -4

Combine like terms: 4 + -4 = 0
-3 + 3x + 2x2 = 0

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

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

Move the constant term to the right:

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

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

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

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

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 = 1.5 + 0.5625

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

Combine like terms: 1.5 + 0.5625 = 2.0625
0.5625 + 1.5x + x2 = 2.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(x + 0.75) equal to 1.436140662 and -1.436140662.

Subproblem 1

x + 0.75 = 1.436140662 Simplifying x + 0.75 = 1.436140662 Reorder the terms: 0.75 + x = 1.436140662 Solving 0.75 + x = 1.436140662 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 = 1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 1.436140662 + -0.75 x = 1.436140662 + -0.75 Combine like terms: 1.436140662 + -0.75 = 0.686140662 x = 0.686140662 Simplifying x = 0.686140662

Subproblem 2

x + 0.75 = -1.436140662 Simplifying x + 0.75 = -1.436140662 Reorder the terms: 0.75 + x = -1.436140662 Solving 0.75 + x = -1.436140662 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 = -1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -1.436140662 + -0.75 x = -1.436140662 + -0.75 Combine like terms: -1.436140662 + -0.75 = -2.186140662 x = -2.186140662 Simplifying x = -2.186140662

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.686140662, -2.186140662}

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