15x+x^2-24=180

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Solution for 15x+x^2-24=180 equation:


Simplifying
15x + x2 + -24 = 180

Reorder the terms:
-24 + 15x + x2 = 180

Solving
-24 + 15x + x2 = 180

Solving for variable 'x'.

Reorder the terms:
-24 + -180 + 15x + x2 = 180 + -180

Combine like terms: -24 + -180 = -204
-204 + 15x + x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-204 + 15x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '204' to each side of the equation.
-204 + 15x + 204 + x2 = 0 + 204

Reorder the terms:
-204 + 204 + 15x + x2 = 0 + 204

Combine like terms: -204 + 204 = 0
0 + 15x + x2 = 0 + 204
15x + x2 = 0 + 204

Combine like terms: 0 + 204 = 204
15x + x2 = 204

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 = 204 + 56.25

Reorder the terms:
56.25 + 15x + x2 = 204 + 56.25

Combine like terms: 204 + 56.25 = 260.25
56.25 + 15x + x2 = 260.25

Factor a perfect square on the left side:
(x + 7.5)(x + 7.5) = 260.25

Calculate the square root of the right side: 16.132265805

Break this problem into two subproblems by setting 
(x + 7.5) equal to 16.132265805 and -16.132265805.

Subproblem 1

x + 7.5 = 16.132265805 Simplifying x + 7.5 = 16.132265805 Reorder the terms: 7.5 + x = 16.132265805 Solving 7.5 + x = 16.132265805 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 = 16.132265805 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = 16.132265805 + -7.5 x = 16.132265805 + -7.5 Combine like terms: 16.132265805 + -7.5 = 8.632265805 x = 8.632265805 Simplifying x = 8.632265805

Subproblem 2

x + 7.5 = -16.132265805 Simplifying x + 7.5 = -16.132265805 Reorder the terms: 7.5 + x = -16.132265805 Solving 7.5 + x = -16.132265805 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 = -16.132265805 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = -16.132265805 + -7.5 x = -16.132265805 + -7.5 Combine like terms: -16.132265805 + -7.5 = -23.632265805 x = -23.632265805 Simplifying x = -23.632265805

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.632265805, -23.632265805}

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