1.5x^2-4.5x=3

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Solution for 1.5x^2-4.5x=3 equation:


Simplifying
1.5x2 + -4.5x = 3

Reorder the terms:
-4.5x + 1.5x2 = 3

Solving
-4.5x + 1.5x2 = 3

Solving for variable 'x'.

Reorder the terms:
-3 + -4.5x + 1.5x2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + -4.5x + 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'.
-2 + -3x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -3x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -3x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -3x + x2 = 0 + 2
-3x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-3x + x2 = 2

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 2 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 2 + 2.25

Combine like terms: 2 + 2.25 = 4.25
2.25 + -3x + x2 = 4.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(x + -1.5) equal to 2.061552813 and -2.061552813.

Subproblem 1

x + -1.5 = 2.061552813 Simplifying x + -1.5 = 2.061552813 Reorder the terms: -1.5 + x = 2.061552813 Solving -1.5 + x = 2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 2.061552813 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 2.061552813 + 1.5 x = 2.061552813 + 1.5 Combine like terms: 2.061552813 + 1.5 = 3.561552813 x = 3.561552813 Simplifying x = 3.561552813

Subproblem 2

x + -1.5 = -2.061552813 Simplifying x + -1.5 = -2.061552813 Reorder the terms: -1.5 + x = -2.061552813 Solving -1.5 + x = -2.061552813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -2.061552813 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -2.061552813 + 1.5 x = -2.061552813 + 1.5 Combine like terms: -2.061552813 + 1.5 = -0.561552813 x = -0.561552813 Simplifying x = -0.561552813

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.561552813, -0.561552813}

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