3=-0.5x^2+1.5x+2

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Solution for 3=-0.5x^2+1.5x+2 equation:


Simplifying
3 = -0.5x2 + 1.5x + 2

Reorder the terms:
3 = 2 + 1.5x + -0.5x2

Solving
3 = 2 + 1.5x + -0.5x2

Solving for variable 'x'.

Combine like terms: 3 + -2 = 1
1 + -1.5x + 0.5x2 = 2 + 1.5x + -0.5x2 + -2 + -1.5x + 0.5x2

Reorder the terms:
1 + -1.5x + 0.5x2 = 2 + -2 + 1.5x + -1.5x + -0.5x2 + 0.5x2

Combine like terms: 2 + -2 = 0
1 + -1.5x + 0.5x2 = 0 + 1.5x + -1.5x + -0.5x2 + 0.5x2
1 + -1.5x + 0.5x2 = 1.5x + -1.5x + -0.5x2 + 0.5x2

Combine like terms: 1.5x + -1.5x = 0.0
1 + -1.5x + 0.5x2 = 0.0 + -0.5x2 + 0.5x2
1 + -1.5x + 0.5x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
1 + -1.5x + 0.5x2 = 0.0

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

Divide each side by '0.5'.
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 = 0.25
2.25 + -3x + x2 = 0.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 0.25

Calculate the square root of the right side: 0.5

Break this problem into two subproblems by setting 
(x + -1.5) equal to 0.5 and -0.5.

Subproblem 1

x + -1.5 = 0.5 Simplifying x + -1.5 = 0.5 Reorder the terms: -1.5 + x = 0.5 Solving -1.5 + x = 0.5 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 = 0.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 0.5 + 1.5 x = 0.5 + 1.5 Combine like terms: 0.5 + 1.5 = 2 x = 2 Simplifying x = 2

Subproblem 2

x + -1.5 = -0.5 Simplifying x + -1.5 = -0.5 Reorder the terms: -1.5 + x = -0.5 Solving -1.5 + x = -0.5 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 = -0.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -0.5 + 1.5 x = -0.5 + 1.5 Combine like terms: -0.5 + 1.5 = 1 x = 1 Simplifying x = 1

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2, 1}

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