9x^2+8x-27=0

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Solution for 9x^2+8x-27=0 equation:


Simplifying
9x2 + 8x + -27 = 0

Reorder the terms:
-27 + 8x + 9x2 = 0

Solving
-27 + 8x + 9x2 = 0

Solving for variable 'x'.

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

Divide each side by '9'.
-3 + 0.8888888889x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 0.8888888889x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + 0.8888888889x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 0.8888888889x + x2 = 0 + 3
0.8888888889x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
0.8888888889x + x2 = 3

The x term is 0.8888888889x.  Take half its coefficient (0.4444444445).
Square it (0.1975308642) and add it to both sides.

Add '0.1975308642' to each side of the equation.
0.8888888889x + 0.1975308642 + x2 = 3 + 0.1975308642

Reorder the terms:
0.1975308642 + 0.8888888889x + x2 = 3 + 0.1975308642

Combine like terms: 3 + 0.1975308642 = 3.1975308642
0.1975308642 + 0.8888888889x + x2 = 3.1975308642

Factor a perfect square on the left side:
(x + 0.4444444445)(x + 0.4444444445) = 3.1975308642

Calculate the square root of the right side: 1.788164104

Break this problem into two subproblems by setting 
(x + 0.4444444445) equal to 1.788164104 and -1.788164104.

Subproblem 1

x + 0.4444444445 = 1.788164104 Simplifying x + 0.4444444445 = 1.788164104 Reorder the terms: 0.4444444445 + x = 1.788164104 Solving 0.4444444445 + x = 1.788164104 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4444444445' to each side of the equation. 0.4444444445 + -0.4444444445 + x = 1.788164104 + -0.4444444445 Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000 0.0000000000 + x = 1.788164104 + -0.4444444445 x = 1.788164104 + -0.4444444445 Combine like terms: 1.788164104 + -0.4444444445 = 1.3437196595 x = 1.3437196595 Simplifying x = 1.3437196595

Subproblem 2

x + 0.4444444445 = -1.788164104 Simplifying x + 0.4444444445 = -1.788164104 Reorder the terms: 0.4444444445 + x = -1.788164104 Solving 0.4444444445 + x = -1.788164104 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4444444445' to each side of the equation. 0.4444444445 + -0.4444444445 + x = -1.788164104 + -0.4444444445 Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000 0.0000000000 + x = -1.788164104 + -0.4444444445 x = -1.788164104 + -0.4444444445 Combine like terms: -1.788164104 + -0.4444444445 = -2.2326085485 x = -2.2326085485 Simplifying x = -2.2326085485

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3437196595, -2.2326085485}

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