.5x^2+3x=33

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Solution for .5x^2+3x=33 equation:


Simplifying
0.5x2 + 3x = 33

Reorder the terms:
3x + 0.5x2 = 33

Solving
3x + 0.5x2 = 33

Solving for variable 'x'.

Reorder the terms:
-33 + 3x + 0.5x2 = 33 + -33

Combine like terms: 33 + -33 = 0
-33 + 3x + 0.5x2 = 0

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

Divide each side by '0.5'.
-66 + 6x + x2 = 0

Move the constant term to the right:

Add '66' to each side of the equation.
-66 + 6x + 66 + x2 = 0 + 66

Reorder the terms:
-66 + 66 + 6x + x2 = 0 + 66

Combine like terms: -66 + 66 = 0
0 + 6x + x2 = 0 + 66
6x + x2 = 0 + 66

Combine like terms: 0 + 66 = 66
6x + x2 = 66

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = 66 + 9

Reorder the terms:
9 + 6x + x2 = 66 + 9

Combine like terms: 66 + 9 = 75
9 + 6x + x2 = 75

Factor a perfect square on the left side:
(x + 3)(x + 3) = 75

Calculate the square root of the right side: 8.660254038

Break this problem into two subproblems by setting 
(x + 3) equal to 8.660254038 and -8.660254038.

Subproblem 1

x + 3 = 8.660254038 Simplifying x + 3 = 8.660254038 Reorder the terms: 3 + x = 8.660254038 Solving 3 + x = 8.660254038 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = 8.660254038 + -3 Combine like terms: 3 + -3 = 0 0 + x = 8.660254038 + -3 x = 8.660254038 + -3 Combine like terms: 8.660254038 + -3 = 5.660254038 x = 5.660254038 Simplifying x = 5.660254038

Subproblem 2

x + 3 = -8.660254038 Simplifying x + 3 = -8.660254038 Reorder the terms: 3 + x = -8.660254038 Solving 3 + x = -8.660254038 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = -8.660254038 + -3 Combine like terms: 3 + -3 = 0 0 + x = -8.660254038 + -3 x = -8.660254038 + -3 Combine like terms: -8.660254038 + -3 = -11.660254038 x = -11.660254038 Simplifying x = -11.660254038

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.660254038, -11.660254038}

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