x^2=-15x-23

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Solution for x^2=-15x-23 equation:


Simplifying
x2 = -15x + -23

Reorder the terms:
x2 = -23 + -15x

Solving
x2 = -23 + -15x

Solving for variable 'x'.

Reorder the terms:
23 + 15x + x2 = -23 + -15x + 23 + 15x

Reorder the terms:
23 + 15x + x2 = -23 + 23 + -15x + 15x

Combine like terms: -23 + 23 = 0
23 + 15x + x2 = 0 + -15x + 15x
23 + 15x + x2 = -15x + 15x

Combine like terms: -15x + 15x = 0
23 + 15x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-23' to each side of the equation.
23 + 15x + -23 + x2 = 0 + -23

Reorder the terms:
23 + -23 + 15x + x2 = 0 + -23

Combine like terms: 23 + -23 = 0
0 + 15x + x2 = 0 + -23
15x + x2 = 0 + -23

Combine like terms: 0 + -23 = -23
15x + x2 = -23

The x term is 15x.  Take half its coefficient (7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
15x + 56.25 + x2 = -23 + 56.25

Reorder the terms:
56.25 + 15x + x2 = -23 + 56.25

Combine like terms: -23 + 56.25 = 33.25
56.25 + 15x + x2 = 33.25

Factor a perfect square on the left side:
(x + 7.5)(x + 7.5) = 33.25

Calculate the square root of the right side: 5.766281297

Break this problem into two subproblems by setting 
(x + 7.5) equal to 5.766281297 and -5.766281297.

Subproblem 1

x + 7.5 = 5.766281297 Simplifying x + 7.5 = 5.766281297 Reorder the terms: 7.5 + x = 5.766281297 Solving 7.5 + x = 5.766281297 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = 5.766281297 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = 5.766281297 + -7.5 x = 5.766281297 + -7.5 Combine like terms: 5.766281297 + -7.5 = -1.733718703 x = -1.733718703 Simplifying x = -1.733718703

Subproblem 2

x + 7.5 = -5.766281297 Simplifying x + 7.5 = -5.766281297 Reorder the terms: 7.5 + x = -5.766281297 Solving 7.5 + x = -5.766281297 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = -5.766281297 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = -5.766281297 + -7.5 x = -5.766281297 + -7.5 Combine like terms: -5.766281297 + -7.5 = -13.266281297 x = -13.266281297 Simplifying x = -13.266281297

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.733718703, -13.266281297}

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