-6x^2+9x+8=0

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Solution for -6x^2+9x+8=0 equation:


Simplifying
-6x2 + 9x + 8 = 0

Reorder the terms:
8 + 9x + -6x2 = 0

Solving
8 + 9x + -6x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
-1.333333333 + -1.5x + x2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -1.5x + 1.333333333 + x2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -1.5x + x2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -1.5x + x2 = 0 + 1.333333333
-1.5x + x2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-1.5x + x2 = 1.333333333

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 = 1.333333333 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 1.333333333 + 0.5625

Combine like terms: 1.333333333 + 0.5625 = 1.895833333
0.5625 + -1.5x + x2 = 1.895833333

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 1.895833333

Calculate the square root of the right side: 1.376892637

Break this problem into two subproblems by setting 
(x + -0.75) equal to 1.376892637 and -1.376892637.

Subproblem 1

x + -0.75 = 1.376892637 Simplifying x + -0.75 = 1.376892637 Reorder the terms: -0.75 + x = 1.376892637 Solving -0.75 + x = 1.376892637 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 = 1.376892637 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.376892637 + 0.75 x = 1.376892637 + 0.75 Combine like terms: 1.376892637 + 0.75 = 2.126892637 x = 2.126892637 Simplifying x = 2.126892637

Subproblem 2

x + -0.75 = -1.376892637 Simplifying x + -0.75 = -1.376892637 Reorder the terms: -0.75 + x = -1.376892637 Solving -0.75 + x = -1.376892637 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 = -1.376892637 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.376892637 + 0.75 x = -1.376892637 + 0.75 Combine like terms: -1.376892637 + 0.75 = -0.626892637 x = -0.626892637 Simplifying x = -0.626892637

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.126892637, -0.626892637}

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