2x^2-15x+117+x^2-8x=180

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Solution for 2x^2-15x+117+x^2-8x=180 equation:


Simplifying
2x2 + -15x + 117 + x2 + -8x = 180

Reorder the terms:
117 + -15x + -8x + 2x2 + x2 = 180

Combine like terms: -15x + -8x = -23x
117 + -23x + 2x2 + x2 = 180

Combine like terms: 2x2 + x2 = 3x2
117 + -23x + 3x2 = 180

Solving
117 + -23x + 3x2 = 180

Solving for variable 'x'.

Reorder the terms:
117 + -180 + -23x + 3x2 = 180 + -180

Combine like terms: 117 + -180 = -63
-63 + -23x + 3x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-63 + -23x + 3x2 = 0

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

Divide each side by '3'.
-21 + -7.666666667x + x2 = 0

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + -7.666666667x + 21 + x2 = 0 + 21

Reorder the terms:
-21 + 21 + -7.666666667x + x2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + -7.666666667x + x2 = 0 + 21
-7.666666667x + x2 = 0 + 21

Combine like terms: 0 + 21 = 21
-7.666666667x + x2 = 21

The x term is -7.666666667x.  Take half its coefficient (-3.833333334).
Square it (14.69444445) and add it to both sides.

Add '14.69444445' to each side of the equation.
-7.666666667x + 14.69444445 + x2 = 21 + 14.69444445

Reorder the terms:
14.69444445 + -7.666666667x + x2 = 21 + 14.69444445

Combine like terms: 21 + 14.69444445 = 35.69444445
14.69444445 + -7.666666667x + x2 = 35.69444445

Factor a perfect square on the left side:
(x + -3.833333334)(x + -3.833333334) = 35.69444445

Calculate the square root of the right side: 5.974482777

Break this problem into two subproblems by setting 
(x + -3.833333334) equal to 5.974482777 and -5.974482777.

Subproblem 1

x + -3.833333334 = 5.974482777 Simplifying x + -3.833333334 = 5.974482777 Reorder the terms: -3.833333334 + x = 5.974482777 Solving -3.833333334 + x = 5.974482777 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.833333334' to each side of the equation. -3.833333334 + 3.833333334 + x = 5.974482777 + 3.833333334 Combine like terms: -3.833333334 + 3.833333334 = 0.000000000 0.000000000 + x = 5.974482777 + 3.833333334 x = 5.974482777 + 3.833333334 Combine like terms: 5.974482777 + 3.833333334 = 9.807816111 x = 9.807816111 Simplifying x = 9.807816111

Subproblem 2

x + -3.833333334 = -5.974482777 Simplifying x + -3.833333334 = -5.974482777 Reorder the terms: -3.833333334 + x = -5.974482777 Solving -3.833333334 + x = -5.974482777 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.833333334' to each side of the equation. -3.833333334 + 3.833333334 + x = -5.974482777 + 3.833333334 Combine like terms: -3.833333334 + 3.833333334 = 0.000000000 0.000000000 + x = -5.974482777 + 3.833333334 x = -5.974482777 + 3.833333334 Combine like terms: -5.974482777 + 3.833333334 = -2.141149443 x = -2.141149443 Simplifying x = -2.141149443

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.807816111, -2.141149443}

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