3x(15-x)=65

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Solution for 3x(15-x)=65 equation:


Simplifying
3x(15 + -1x) = 65
(15 * 3x + -1x * 3x) = 65
(45x + -3x2) = 65

Solving
45x + -3x2 = 65

Solving for variable 'x'.

Reorder the terms:
-65 + 45x + -3x2 = 65 + -65

Combine like terms: 65 + -65 = 0
-65 + 45x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
21.66666667 + -15x + x2 = 0

Move the constant term to the right:

Add '-21.66666667' to each side of the equation.
21.66666667 + -15x + -21.66666667 + x2 = 0 + -21.66666667

Reorder the terms:
21.66666667 + -21.66666667 + -15x + x2 = 0 + -21.66666667

Combine like terms: 21.66666667 + -21.66666667 = 0.00000000
0.00000000 + -15x + x2 = 0 + -21.66666667
-15x + x2 = 0 + -21.66666667

Combine like terms: 0 + -21.66666667 = -21.66666667
-15x + x2 = -21.66666667

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 = -21.66666667 + 56.25

Reorder the terms:
56.25 + -15x + x2 = -21.66666667 + 56.25

Combine like terms: -21.66666667 + 56.25 = 34.58333333
56.25 + -15x + x2 = 34.58333333

Factor a perfect square on the left side:
(x + -7.5)(x + -7.5) = 34.58333333

Calculate the square root of the right side: 5.880759588

Break this problem into two subproblems by setting 
(x + -7.5) equal to 5.880759588 and -5.880759588.

Subproblem 1

x + -7.5 = 5.880759588 Simplifying x + -7.5 = 5.880759588 Reorder the terms: -7.5 + x = 5.880759588 Solving -7.5 + x = 5.880759588 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 = 5.880759588 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = 5.880759588 + 7.5 x = 5.880759588 + 7.5 Combine like terms: 5.880759588 + 7.5 = 13.380759588 x = 13.380759588 Simplifying x = 13.380759588

Subproblem 2

x + -7.5 = -5.880759588 Simplifying x + -7.5 = -5.880759588 Reorder the terms: -7.5 + x = -5.880759588 Solving -7.5 + x = -5.880759588 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 = -5.880759588 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = -5.880759588 + 7.5 x = -5.880759588 + 7.5 Combine like terms: -5.880759588 + 7.5 = 1.619240412 x = 1.619240412 Simplifying x = 1.619240412

Solution

The solution to the problem is based on the solutions from the subproblems. x = {13.380759588, 1.619240412}

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