x^2+5x+24=180

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Solution for x^2+5x+24=180 equation:


Simplifying
x2 + 5x + 24 = 180

Reorder the terms:
24 + 5x + x2 = 180

Solving
24 + 5x + x2 = 180

Solving for variable 'x'.

Reorder the terms:
24 + -180 + 5x + x2 = 180 + -180

Combine like terms: 24 + -180 = -156
-156 + 5x + x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-156 + 5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '156' to each side of the equation.
-156 + 5x + 156 + x2 = 0 + 156

Reorder the terms:
-156 + 156 + 5x + x2 = 0 + 156

Combine like terms: -156 + 156 = 0
0 + 5x + x2 = 0 + 156
5x + x2 = 0 + 156

Combine like terms: 0 + 156 = 156
5x + x2 = 156

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 = 156 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 156 + 6.25

Combine like terms: 156 + 6.25 = 162.25
6.25 + 5x + x2 = 162.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 162.25

Calculate the square root of the right side: 12.737739203

Break this problem into two subproblems by setting 
(x + 2.5) equal to 12.737739203 and -12.737739203.

Subproblem 1

x + 2.5 = 12.737739203 Simplifying x + 2.5 = 12.737739203 Reorder the terms: 2.5 + x = 12.737739203 Solving 2.5 + x = 12.737739203 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 = 12.737739203 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 12.737739203 + -2.5 x = 12.737739203 + -2.5 Combine like terms: 12.737739203 + -2.5 = 10.237739203 x = 10.237739203 Simplifying x = 10.237739203

Subproblem 2

x + 2.5 = -12.737739203 Simplifying x + 2.5 = -12.737739203 Reorder the terms: 2.5 + x = -12.737739203 Solving 2.5 + x = -12.737739203 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 = -12.737739203 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -12.737739203 + -2.5 x = -12.737739203 + -2.5 Combine like terms: -12.737739203 + -2.5 = -15.237739203 x = -15.237739203 Simplifying x = -15.237739203

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10.237739203, -15.237739203}

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