5x-3=19-x^2

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Solution for 5x-3=19-x^2 equation:


Simplifying
5x + -3 = 19 + -1x2

Reorder the terms:
-3 + 5x = 19 + -1x2

Solving
-3 + 5x = 19 + -1x2

Solving for variable 'x'.

Reorder the terms:
-3 + -19 + 5x + x2 = 19 + -1x2 + -19 + x2

Combine like terms: -3 + -19 = -22
-22 + 5x + x2 = 19 + -1x2 + -19 + x2

Reorder the terms:
-22 + 5x + x2 = 19 + -19 + -1x2 + x2

Combine like terms: 19 + -19 = 0
-22 + 5x + x2 = 0 + -1x2 + x2
-22 + 5x + x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
-22 + 5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '22' to each side of the equation.
-22 + 5x + 22 + x2 = 0 + 22

Reorder the terms:
-22 + 22 + 5x + x2 = 0 + 22

Combine like terms: -22 + 22 = 0
0 + 5x + x2 = 0 + 22
5x + x2 = 0 + 22

Combine like terms: 0 + 22 = 22
5x + x2 = 22

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = 22 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 22 + 6.25

Combine like terms: 22 + 6.25 = 28.25
6.25 + 5x + x2 = 28.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 28.25

Calculate the square root of the right side: 5.315072906

Break this problem into two subproblems by setting 
(x + 2.5) equal to 5.315072906 and -5.315072906.

Subproblem 1

x + 2.5 = 5.315072906 Simplifying x + 2.5 = 5.315072906 Reorder the terms: 2.5 + x = 5.315072906 Solving 2.5 + x = 5.315072906 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 5.315072906 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 5.315072906 + -2.5 x = 5.315072906 + -2.5 Combine like terms: 5.315072906 + -2.5 = 2.815072906 x = 2.815072906 Simplifying x = 2.815072906

Subproblem 2

x + 2.5 = -5.315072906 Simplifying x + 2.5 = -5.315072906 Reorder the terms: 2.5 + x = -5.315072906 Solving 2.5 + x = -5.315072906 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -5.315072906 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -5.315072906 + -2.5 x = -5.315072906 + -2.5 Combine like terms: -5.315072906 + -2.5 = -7.815072906 x = -7.815072906 Simplifying x = -7.815072906

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.815072906, -7.815072906}

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