x^2+15x=65

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Solution for x^2+15x=65 equation:


Simplifying
x2 + 15x = 65

Reorder the terms:
15x + x2 = 65

Solving
15x + x2 = 65

Solving for variable 'x'.

Reorder the terms:
-65 + 15x + x2 = 65 + -65

Combine like terms: 65 + -65 = 0
-65 + 15x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '65' to each side of the equation.
-65 + 15x + 65 + x2 = 0 + 65

Reorder the terms:
-65 + 65 + 15x + x2 = 0 + 65

Combine like terms: -65 + 65 = 0
0 + 15x + x2 = 0 + 65
15x + x2 = 0 + 65

Combine like terms: 0 + 65 = 65
15x + x2 = 65

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 = 65 + 56.25

Reorder the terms:
56.25 + 15x + x2 = 65 + 56.25

Combine like terms: 65 + 56.25 = 121.25
56.25 + 15x + x2 = 121.25

Factor a perfect square on the left side:
(x + 7.5)(x + 7.5) = 121.25

Calculate the square root of the right side: 11.011357773

Break this problem into two subproblems by setting 
(x + 7.5) equal to 11.011357773 and -11.011357773.

Subproblem 1

x + 7.5 = 11.011357773 Simplifying x + 7.5 = 11.011357773 Reorder the terms: 7.5 + x = 11.011357773 Solving 7.5 + x = 11.011357773 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 = 11.011357773 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = 11.011357773 + -7.5 x = 11.011357773 + -7.5 Combine like terms: 11.011357773 + -7.5 = 3.511357773 x = 3.511357773 Simplifying x = 3.511357773

Subproblem 2

x + 7.5 = -11.011357773 Simplifying x + 7.5 = -11.011357773 Reorder the terms: 7.5 + x = -11.011357773 Solving 7.5 + x = -11.011357773 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 = -11.011357773 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = -11.011357773 + -7.5 x = -11.011357773 + -7.5 Combine like terms: -11.011357773 + -7.5 = -18.511357773 x = -18.511357773 Simplifying x = -18.511357773

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.511357773, -18.511357773}

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