x^2+3x=22

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Solution for x^2+3x=22 equation:


Simplifying
x2 + 3x = 22

Reorder the terms:
3x + x2 = 22

Solving
3x + x2 = 22

Solving for variable 'x'.

Reorder the terms:
-22 + 3x + x2 = 22 + -22

Combine like terms: 22 + -22 = 0
-22 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '22' to each side of the equation.
-22 + 3x + 22 + x2 = 0 + 22

Reorder the terms:
-22 + 22 + 3x + x2 = 0 + 22

Combine like terms: -22 + 22 = 0
0 + 3x + x2 = 0 + 22
3x + x2 = 0 + 22

Combine like terms: 0 + 22 = 22
3x + x2 = 22

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 = 22 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 22 + 2.25

Combine like terms: 22 + 2.25 = 24.25
2.25 + 3x + x2 = 24.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 24.25

Calculate the square root of the right side: 4.924428901

Break this problem into two subproblems by setting 
(x + 1.5) equal to 4.924428901 and -4.924428901.

Subproblem 1

x + 1.5 = 4.924428901 Simplifying x + 1.5 = 4.924428901 Reorder the terms: 1.5 + x = 4.924428901 Solving 1.5 + x = 4.924428901 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 = 4.924428901 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 4.924428901 + -1.5 x = 4.924428901 + -1.5 Combine like terms: 4.924428901 + -1.5 = 3.424428901 x = 3.424428901 Simplifying x = 3.424428901

Subproblem 2

x + 1.5 = -4.924428901 Simplifying x + 1.5 = -4.924428901 Reorder the terms: 1.5 + x = -4.924428901 Solving 1.5 + x = -4.924428901 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 = -4.924428901 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -4.924428901 + -1.5 x = -4.924428901 + -1.5 Combine like terms: -4.924428901 + -1.5 = -6.424428901 x = -6.424428901 Simplifying x = -6.424428901

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.424428901, -6.424428901}

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