1.5x^2+6x=7.5

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Solution for 1.5x^2+6x=7.5 equation:


Simplifying
1.5x2 + 6x = 7.5

Reorder the terms:
6x + 1.5x2 = 7.5

Solving
6x + 1.5x2 = 7.5

Solving for variable 'x'.

Reorder the terms:
-7.5 + 6x + 1.5x2 = 7.5 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
-7.5 + 6x + 1.5x2 = 0.0

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

Divide each side by '1.5'.
-5 + 4x + x2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 4x + 5 + x2 = 0 + 5

Reorder the terms:
-5 + 5 + 4x + x2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 4x + x2 = 0 + 5
4x + x2 = 0 + 5

Combine like terms: 0 + 5 = 5
4x + x2 = 5

The x term is 4x.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4x + 4 + x2 = 5 + 4

Reorder the terms:
4 + 4x + x2 = 5 + 4

Combine like terms: 5 + 4 = 9
4 + 4x + x2 = 9

Factor a perfect square on the left side:
(x + 2)(x + 2) = 9

Calculate the square root of the right side: 3

Break this problem into two subproblems by setting 
(x + 2) equal to 3 and -3.

Subproblem 1

x + 2 = 3 Simplifying x + 2 = 3 Reorder the terms: 2 + x = 3 Solving 2 + x = 3 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = 3 + -2 Combine like terms: 2 + -2 = 0 0 + x = 3 + -2 x = 3 + -2 Combine like terms: 3 + -2 = 1 x = 1 Simplifying x = 1

Subproblem 2

x + 2 = -3 Simplifying x + 2 = -3 Reorder the terms: 2 + x = -3 Solving 2 + x = -3 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + x = -3 + -2 Combine like terms: 2 + -2 = 0 0 + x = -3 + -2 x = -3 + -2 Combine like terms: -3 + -2 = -5 x = -5 Simplifying x = -5

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1, -5}

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