x*1.5x+3x=120

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Solution for x*1.5x+3x=120 equation:


Simplifying
x * 1.5x + 3x = 120

Reorder the terms for easier multiplication:
1.5x * x + 3x = 120

Multiply x * x
1.5x2 + 3x = 120

Reorder the terms:
3x + 1.5x2 = 120

Solving
3x + 1.5x2 = 120

Solving for variable 'x'.

Reorder the terms:
-120 + 3x + 1.5x2 = 120 + -120

Combine like terms: 120 + -120 = 0
-120 + 3x + 1.5x2 = 0

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

Divide each side by '1.5'.
-80 + 2x + x2 = 0

Move the constant term to the right:

Add '80' to each side of the equation.
-80 + 2x + 80 + x2 = 0 + 80

Reorder the terms:
-80 + 80 + 2x + x2 = 0 + 80

Combine like terms: -80 + 80 = 0
0 + 2x + x2 = 0 + 80
2x + x2 = 0 + 80

Combine like terms: 0 + 80 = 80
2x + x2 = 80

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 80 + 1

Reorder the terms:
1 + 2x + x2 = 80 + 1

Combine like terms: 80 + 1 = 81
1 + 2x + x2 = 81

Factor a perfect square on the left side:
(x + 1)(x + 1) = 81

Calculate the square root of the right side: 9

Break this problem into two subproblems by setting 
(x + 1) equal to 9 and -9.

Subproblem 1

x + 1 = 9 Simplifying x + 1 = 9 Reorder the terms: 1 + x = 9 Solving 1 + x = 9 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 9 + -1 Combine like terms: 1 + -1 = 0 0 + x = 9 + -1 x = 9 + -1 Combine like terms: 9 + -1 = 8 x = 8 Simplifying x = 8

Subproblem 2

x + 1 = -9 Simplifying x + 1 = -9 Reorder the terms: 1 + x = -9 Solving 1 + x = -9 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -9 + -1 Combine like terms: 1 + -1 = 0 0 + x = -9 + -1 x = -9 + -1 Combine like terms: -9 + -1 = -10 x = -10 Simplifying x = -10

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8, -10}

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