x^2-19=-3x

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Solution for x^2-19=-3x equation:


Simplifying
x2 + -19 = -3x

Reorder the terms:
-19 + x2 = -3x

Solving
-19 + x2 = -3x

Solving for variable 'x'.

Reorder the terms:
-19 + 3x + x2 = -3x + 3x

Combine like terms: -3x + 3x = 0
-19 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + 3x + 19 + x2 = 0 + 19

Reorder the terms:
-19 + 19 + 3x + x2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + 3x + x2 = 0 + 19
3x + x2 = 0 + 19

Combine like terms: 0 + 19 = 19
3x + x2 = 19

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 = 19 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 19 + 2.25

Combine like terms: 19 + 2.25 = 21.25
2.25 + 3x + x2 = 21.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(x + 1.5) equal to 4.609772229 and -4.609772229.

Subproblem 1

x + 1.5 = 4.609772229 Simplifying x + 1.5 = 4.609772229 Reorder the terms: 1.5 + x = 4.609772229 Solving 1.5 + x = 4.609772229 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 = 4.609772229 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 4.609772229 + -1.5 x = 4.609772229 + -1.5 Combine like terms: 4.609772229 + -1.5 = 3.109772229 x = 3.109772229 Simplifying x = 3.109772229

Subproblem 2

x + 1.5 = -4.609772229 Simplifying x + 1.5 = -4.609772229 Reorder the terms: 1.5 + x = -4.609772229 Solving 1.5 + x = -4.609772229 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 = -4.609772229 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -4.609772229 + -1.5 x = -4.609772229 + -1.5 Combine like terms: -4.609772229 + -1.5 = -6.109772229 x = -6.109772229 Simplifying x = -6.109772229

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.109772229, -6.109772229}

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