=(x^2+1.5x-6)

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Solution for =(x^2+1.5x-6) equation:


Simplifying
0 = (x2 + 1.5x + -6)

Reorder the terms:
0 = (-6 + 1.5x + x2)

Remove parenthesis around (-6 + 1.5x + x2)
0 = -6 + 1.5x + x2

Solving
0 = -6 + 1.5x + x2

Solving for variable 'x'.

Combine like terms: 0 + 6 = 6
6 + -1.5x + -1x2 = -6 + 1.5x + x2 + 6 + -1.5x + -1x2

Reorder the terms:
6 + -1.5x + -1x2 = -6 + 6 + 1.5x + -1.5x + x2 + -1x2

Combine like terms: -6 + 6 = 0
6 + -1.5x + -1x2 = 0 + 1.5x + -1.5x + x2 + -1x2
6 + -1.5x + -1x2 = 1.5x + -1.5x + x2 + -1x2

Combine like terms: 1.5x + -1.5x = 0.0
6 + -1.5x + -1x2 = 0.0 + x2 + -1x2
6 + -1.5x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
6 + -1.5x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-6 + 1.5x + x2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 1.5x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + 1.5x + x2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 1.5x + x2 = 0 + 6
1.5x + x2 = 0 + 6

Combine like terms: 0 + 6 = 6
1.5x + x2 = 6

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 6 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 6 + 0.5625

Combine like terms: 6 + 0.5625 = 6.5625
0.5625 + 1.5x + x2 = 6.5625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 6.5625

Calculate the square root of the right side: 2.561737691

Break this problem into two subproblems by setting 
(x + 0.75) equal to 2.561737691 and -2.561737691.

Subproblem 1

x + 0.75 = 2.561737691 Simplifying x + 0.75 = 2.561737691 Reorder the terms: 0.75 + x = 2.561737691 Solving 0.75 + x = 2.561737691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 2.561737691 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 2.561737691 + -0.75 x = 2.561737691 + -0.75 Combine like terms: 2.561737691 + -0.75 = 1.811737691 x = 1.811737691 Simplifying x = 1.811737691

Subproblem 2

x + 0.75 = -2.561737691 Simplifying x + 0.75 = -2.561737691 Reorder the terms: 0.75 + x = -2.561737691 Solving 0.75 + x = -2.561737691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -2.561737691 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -2.561737691 + -0.75 x = -2.561737691 + -0.75 Combine like terms: -2.561737691 + -0.75 = -3.311737691 x = -3.311737691 Simplifying x = -3.311737691

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.811737691, -3.311737691}

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