.5x^2+1.5x-2=3000

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Solution for .5x^2+1.5x-2=3000 equation:


Simplifying
0.5x2 + 1.5x + -2 = 3000

Reorder the terms:
-2 + 1.5x + 0.5x2 = 3000

Solving
-2 + 1.5x + 0.5x2 = 3000

Solving for variable 'x'.

Reorder the terms:
-2 + -3000 + 1.5x + 0.5x2 = 3000 + -3000

Combine like terms: -2 + -3000 = -3002
-3002 + 1.5x + 0.5x2 = 3000 + -3000

Combine like terms: 3000 + -3000 = 0
-3002 + 1.5x + 0.5x2 = 0

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

Divide each side by '0.5'.
-6004 + 3x + x2 = 0

Move the constant term to the right:

Add '6004' to each side of the equation.
-6004 + 3x + 6004 + x2 = 0 + 6004

Reorder the terms:
-6004 + 6004 + 3x + x2 = 0 + 6004

Combine like terms: -6004 + 6004 = 0
0 + 3x + x2 = 0 + 6004
3x + x2 = 0 + 6004

Combine like terms: 0 + 6004 = 6004
3x + x2 = 6004

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 = 6004 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 6004 + 2.25

Combine like terms: 6004 + 2.25 = 6006.25
2.25 + 3x + x2 = 6006.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 6006.25

Calculate the square root of the right side: 77.5

Break this problem into two subproblems by setting 
(x + 1.5) equal to 77.5 and -77.5.

Subproblem 1

x + 1.5 = 77.5 Simplifying x + 1.5 = 77.5 Reorder the terms: 1.5 + x = 77.5 Solving 1.5 + x = 77.5 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 = 77.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 77.5 + -1.5 x = 77.5 + -1.5 Combine like terms: 77.5 + -1.5 = 76 x = 76 Simplifying x = 76

Subproblem 2

x + 1.5 = -77.5 Simplifying x + 1.5 = -77.5 Reorder the terms: 1.5 + x = -77.5 Solving 1.5 + x = -77.5 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 = -77.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -77.5 + -1.5 x = -77.5 + -1.5 Combine like terms: -77.5 + -1.5 = -79 x = -79 Simplifying x = -79

Solution

The solution to the problem is based on the solutions from the subproblems. x = {76, -79}

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