27x^2+32x=52

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Solution for 27x^2+32x=52 equation:


Simplifying
27x2 + 32x = 52

Reorder the terms:
32x + 27x2 = 52

Solving
32x + 27x2 = 52

Solving for variable 'x'.

Reorder the terms:
-52 + 32x + 27x2 = 52 + -52

Combine like terms: 52 + -52 = 0
-52 + 32x + 27x2 = 0

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

Divide each side by '27'.
-1.925925926 + 1.185185185x + x2 = 0

Move the constant term to the right:

Add '1.925925926' to each side of the equation.
-1.925925926 + 1.185185185x + 1.925925926 + x2 = 0 + 1.925925926

Reorder the terms:
-1.925925926 + 1.925925926 + 1.185185185x + x2 = 0 + 1.925925926

Combine like terms: -1.925925926 + 1.925925926 = 0.000000000
0.000000000 + 1.185185185x + x2 = 0 + 1.925925926
1.185185185x + x2 = 0 + 1.925925926

Combine like terms: 0 + 1.925925926 = 1.925925926
1.185185185x + x2 = 1.925925926

The x term is 1.185185185x.  Take half its coefficient (0.5925925925).
Square it (0.3511659807) and add it to both sides.

Add '0.3511659807' to each side of the equation.
1.185185185x + 0.3511659807 + x2 = 1.925925926 + 0.3511659807

Reorder the terms:
0.3511659807 + 1.185185185x + x2 = 1.925925926 + 0.3511659807

Combine like terms: 1.925925926 + 0.3511659807 = 2.2770919067
0.3511659807 + 1.185185185x + x2 = 2.2770919067

Factor a perfect square on the left side:
(x + 0.5925925925)(x + 0.5925925925) = 2.2770919067

Calculate the square root of the right side: 1.509003614

Break this problem into two subproblems by setting 
(x + 0.5925925925) equal to 1.509003614 and -1.509003614.

Subproblem 1

x + 0.5925925925 = 1.509003614 Simplifying x + 0.5925925925 = 1.509003614 Reorder the terms: 0.5925925925 + x = 1.509003614 Solving 0.5925925925 + x = 1.509003614 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5925925925' to each side of the equation. 0.5925925925 + -0.5925925925 + x = 1.509003614 + -0.5925925925 Combine like terms: 0.5925925925 + -0.5925925925 = 0.0000000000 0.0000000000 + x = 1.509003614 + -0.5925925925 x = 1.509003614 + -0.5925925925 Combine like terms: 1.509003614 + -0.5925925925 = 0.9164110215 x = 0.9164110215 Simplifying x = 0.9164110215

Subproblem 2

x + 0.5925925925 = -1.509003614 Simplifying x + 0.5925925925 = -1.509003614 Reorder the terms: 0.5925925925 + x = -1.509003614 Solving 0.5925925925 + x = -1.509003614 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5925925925' to each side of the equation. 0.5925925925 + -0.5925925925 + x = -1.509003614 + -0.5925925925 Combine like terms: 0.5925925925 + -0.5925925925 = 0.0000000000 0.0000000000 + x = -1.509003614 + -0.5925925925 x = -1.509003614 + -0.5925925925 Combine like terms: -1.509003614 + -0.5925925925 = -2.1015962065 x = -2.1015962065 Simplifying x = -2.1015962065

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.9164110215, -2.1015962065}

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