x^2+15x=78

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Solution for x^2+15x=78 equation:


Simplifying
x2 + 15x = 78

Reorder the terms:
15x + x2 = 78

Solving
15x + x2 = 78

Solving for variable 'x'.

Reorder the terms:
-78 + 15x + x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-78 + 15x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + 15x + 78 + x2 = 0 + 78

Reorder the terms:
-78 + 78 + 15x + x2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + 15x + x2 = 0 + 78
15x + x2 = 0 + 78

Combine like terms: 0 + 78 = 78
15x + x2 = 78

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 = 78 + 56.25

Reorder the terms:
56.25 + 15x + x2 = 78 + 56.25

Combine like terms: 78 + 56.25 = 134.25
56.25 + 15x + x2 = 134.25

Factor a perfect square on the left side:
(x + 7.5)(x + 7.5) = 134.25

Calculate the square root of the right side: 11.586630226

Break this problem into two subproblems by setting 
(x + 7.5) equal to 11.586630226 and -11.586630226.

Subproblem 1

x + 7.5 = 11.586630226 Simplifying x + 7.5 = 11.586630226 Reorder the terms: 7.5 + x = 11.586630226 Solving 7.5 + x = 11.586630226 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 = 11.586630226 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = 11.586630226 + -7.5 x = 11.586630226 + -7.5 Combine like terms: 11.586630226 + -7.5 = 4.086630226 x = 4.086630226 Simplifying x = 4.086630226

Subproblem 2

x + 7.5 = -11.586630226 Simplifying x + 7.5 = -11.586630226 Reorder the terms: 7.5 + x = -11.586630226 Solving 7.5 + x = -11.586630226 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 = -11.586630226 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = -11.586630226 + -7.5 x = -11.586630226 + -7.5 Combine like terms: -11.586630226 + -7.5 = -19.086630226 x = -19.086630226 Simplifying x = -19.086630226

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.086630226, -19.086630226}

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