45=2x^2+3x+2

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Solution for 45=2x^2+3x+2 equation:


Simplifying
45 = 2x2 + 3x + 2

Reorder the terms:
45 = 2 + 3x + 2x2

Solving
45 = 2 + 3x + 2x2

Solving for variable 'x'.

Combine like terms: 45 + -2 = 43
43 + -3x + -2x2 = 2 + 3x + 2x2 + -2 + -3x + -2x2

Reorder the terms:
43 + -3x + -2x2 = 2 + -2 + 3x + -3x + 2x2 + -2x2

Combine like terms: 2 + -2 = 0
43 + -3x + -2x2 = 0 + 3x + -3x + 2x2 + -2x2
43 + -3x + -2x2 = 3x + -3x + 2x2 + -2x2

Combine like terms: 3x + -3x = 0
43 + -3x + -2x2 = 0 + 2x2 + -2x2
43 + -3x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
43 + -3x + -2x2 = 0

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

Divide each side by '-2'.
-21.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '21.5' to each side of the equation.
-21.5 + 1.5x + 21.5 + x2 = 0 + 21.5

Reorder the terms:
-21.5 + 21.5 + 1.5x + x2 = 0 + 21.5

Combine like terms: -21.5 + 21.5 = 0.0
0.0 + 1.5x + x2 = 0 + 21.5
1.5x + x2 = 0 + 21.5

Combine like terms: 0 + 21.5 = 21.5
1.5x + x2 = 21.5

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 = 21.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 21.5 + 0.5625

Combine like terms: 21.5 + 0.5625 = 22.0625
0.5625 + 1.5x + x2 = 22.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 22.0625

Calculate the square root of the right side: 4.697073557

Break this problem into two subproblems by setting 
(x + 0.75) equal to 4.697073557 and -4.697073557.

Subproblem 1

x + 0.75 = 4.697073557 Simplifying x + 0.75 = 4.697073557 Reorder the terms: 0.75 + x = 4.697073557 Solving 0.75 + x = 4.697073557 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 = 4.697073557 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 4.697073557 + -0.75 x = 4.697073557 + -0.75 Combine like terms: 4.697073557 + -0.75 = 3.947073557 x = 3.947073557 Simplifying x = 3.947073557

Subproblem 2

x + 0.75 = -4.697073557 Simplifying x + 0.75 = -4.697073557 Reorder the terms: 0.75 + x = -4.697073557 Solving 0.75 + x = -4.697073557 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 = -4.697073557 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -4.697073557 + -0.75 x = -4.697073557 + -0.75 Combine like terms: -4.697073557 + -0.75 = -5.447073557 x = -5.447073557 Simplifying x = -5.447073557

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.947073557, -5.447073557}

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