x^2+3x=117

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Solution for x^2+3x=117 equation:


Simplifying
x2 + 3x = 117

Reorder the terms:
3x + x2 = 117

Solving
3x + x2 = 117

Solving for variable 'x'.

Reorder the terms:
-117 + 3x + x2 = 117 + -117

Combine like terms: 117 + -117 = 0
-117 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '117' to each side of the equation.
-117 + 3x + 117 + x2 = 0 + 117

Reorder the terms:
-117 + 117 + 3x + x2 = 0 + 117

Combine like terms: -117 + 117 = 0
0 + 3x + x2 = 0 + 117
3x + x2 = 0 + 117

Combine like terms: 0 + 117 = 117
3x + x2 = 117

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 = 117 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 117 + 2.25

Combine like terms: 117 + 2.25 = 119.25
2.25 + 3x + x2 = 119.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 119.25

Calculate the square root of the right side: 10.920164834

Break this problem into two subproblems by setting 
(x + 1.5) equal to 10.920164834 and -10.920164834.

Subproblem 1

x + 1.5 = 10.920164834 Simplifying x + 1.5 = 10.920164834 Reorder the terms: 1.5 + x = 10.920164834 Solving 1.5 + x = 10.920164834 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 = 10.920164834 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 10.920164834 + -1.5 x = 10.920164834 + -1.5 Combine like terms: 10.920164834 + -1.5 = 9.420164834 x = 9.420164834 Simplifying x = 9.420164834

Subproblem 2

x + 1.5 = -10.920164834 Simplifying x + 1.5 = -10.920164834 Reorder the terms: 1.5 + x = -10.920164834 Solving 1.5 + x = -10.920164834 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 = -10.920164834 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -10.920164834 + -1.5 x = -10.920164834 + -1.5 Combine like terms: -10.920164834 + -1.5 = -12.420164834 x = -12.420164834 Simplifying x = -12.420164834

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.420164834, -12.420164834}

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