x^2+3x=312

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Solution for x^2+3x=312 equation:


Simplifying
x2 + 3x = 312

Reorder the terms:
3x + x2 = 312

Solving
3x + x2 = 312

Solving for variable 'x'.

Reorder the terms:
-312 + 3x + x2 = 312 + -312

Combine like terms: 312 + -312 = 0
-312 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '312' to each side of the equation.
-312 + 3x + 312 + x2 = 0 + 312

Reorder the terms:
-312 + 312 + 3x + x2 = 0 + 312

Combine like terms: -312 + 312 = 0
0 + 3x + x2 = 0 + 312
3x + x2 = 0 + 312

Combine like terms: 0 + 312 = 312
3x + x2 = 312

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 312 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 312 + 2.25

Combine like terms: 312 + 2.25 = 314.25
2.25 + 3x + x2 = 314.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 314.25

Calculate the square root of the right side: 17.727097901

Break this problem into two subproblems by setting 
(x + 1.5) equal to 17.727097901 and -17.727097901.

Subproblem 1

x + 1.5 = 17.727097901 Simplifying x + 1.5 = 17.727097901 Reorder the terms: 1.5 + x = 17.727097901 Solving 1.5 + x = 17.727097901 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 17.727097901 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 17.727097901 + -1.5 x = 17.727097901 + -1.5 Combine like terms: 17.727097901 + -1.5 = 16.227097901 x = 16.227097901 Simplifying x = 16.227097901

Subproblem 2

x + 1.5 = -17.727097901 Simplifying x + 1.5 = -17.727097901 Reorder the terms: 1.5 + x = -17.727097901 Solving 1.5 + x = -17.727097901 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -17.727097901 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -17.727097901 + -1.5 x = -17.727097901 + -1.5 Combine like terms: -17.727097901 + -1.5 = -19.227097901 x = -19.227097901 Simplifying x = -19.227097901

Solution

The solution to the problem is based on the solutions from the subproblems. x = {16.227097901, -19.227097901}

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