(2x+3)(x)=37

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Solution for (2x+3)(x)=37 equation:


Simplifying
(2x + 3)(x) = 37

Reorder the terms:
(3 + 2x)(x) = 37

Reorder the terms for easier multiplication:
x(3 + 2x) = 37
(3 * x + 2x * x) = 37
(3x + 2x2) = 37

Solving
3x + 2x2 = 37

Solving for variable 'x'.

Reorder the terms:
-37 + 3x + 2x2 = 37 + -37

Combine like terms: 37 + -37 = 0
-37 + 3x + 2x2 = 0

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

Divide each side by '2'.
-18.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '18.5' to each side of the equation.
-18.5 + 1.5x + 18.5 + x2 = 0 + 18.5

Reorder the terms:
-18.5 + 18.5 + 1.5x + x2 = 0 + 18.5

Combine like terms: -18.5 + 18.5 = 0.0
0.0 + 1.5x + x2 = 0 + 18.5
1.5x + x2 = 0 + 18.5

Combine like terms: 0 + 18.5 = 18.5
1.5x + x2 = 18.5

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 = 18.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 18.5 + 0.5625

Combine like terms: 18.5 + 0.5625 = 19.0625
0.5625 + 1.5x + x2 = 19.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 19.0625

Calculate the square root of the right side: 4.366062299

Break this problem into two subproblems by setting 
(x + 0.75) equal to 4.366062299 and -4.366062299.

Subproblem 1

x + 0.75 = 4.366062299 Simplifying x + 0.75 = 4.366062299 Reorder the terms: 0.75 + x = 4.366062299 Solving 0.75 + x = 4.366062299 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 = 4.366062299 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 4.366062299 + -0.75 x = 4.366062299 + -0.75 Combine like terms: 4.366062299 + -0.75 = 3.616062299 x = 3.616062299 Simplifying x = 3.616062299

Subproblem 2

x + 0.75 = -4.366062299 Simplifying x + 0.75 = -4.366062299 Reorder the terms: 0.75 + x = -4.366062299 Solving 0.75 + x = -4.366062299 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 = -4.366062299 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -4.366062299 + -0.75 x = -4.366062299 + -0.75 Combine like terms: -4.366062299 + -0.75 = -5.116062299 x = -5.116062299 Simplifying x = -5.116062299

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.616062299, -5.116062299}

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