32x^2-9x-3=0

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Solution for 32x^2-9x-3=0 equation:


Simplifying
32x2 + -9x + -3 = 0

Reorder the terms:
-3 + -9x + 32x2 = 0

Solving
-3 + -9x + 32x2 = 0

Solving for variable 'x'.

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

Divide each side by '32'.
-0.09375 + -0.28125x + x2 = 0

Move the constant term to the right:

Add '0.09375' to each side of the equation.
-0.09375 + -0.28125x + 0.09375 + x2 = 0 + 0.09375

Reorder the terms:
-0.09375 + 0.09375 + -0.28125x + x2 = 0 + 0.09375

Combine like terms: -0.09375 + 0.09375 = 0.00000
0.00000 + -0.28125x + x2 = 0 + 0.09375
-0.28125x + x2 = 0 + 0.09375

Combine like terms: 0 + 0.09375 = 0.09375
-0.28125x + x2 = 0.09375

The x term is -0.28125x.  Take half its coefficient (-0.140625).
Square it (0.01977539063) and add it to both sides.

Add '0.01977539063' to each side of the equation.
-0.28125x + 0.01977539063 + x2 = 0.09375 + 0.01977539063

Reorder the terms:
0.01977539063 + -0.28125x + x2 = 0.09375 + 0.01977539063

Combine like terms: 0.09375 + 0.01977539063 = 0.11352539063
0.01977539063 + -0.28125x + x2 = 0.11352539063

Factor a perfect square on the left side:
(x + -0.140625)(x + -0.140625) = 0.11352539063

Calculate the square root of the right side: 0.336935291

Break this problem into two subproblems by setting 
(x + -0.140625) equal to 0.336935291 and -0.336935291.

Subproblem 1

x + -0.140625 = 0.336935291 Simplifying x + -0.140625 = 0.336935291 Reorder the terms: -0.140625 + x = 0.336935291 Solving -0.140625 + x = 0.336935291 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.140625' to each side of the equation. -0.140625 + 0.140625 + x = 0.336935291 + 0.140625 Combine like terms: -0.140625 + 0.140625 = 0.000000 0.000000 + x = 0.336935291 + 0.140625 x = 0.336935291 + 0.140625 Combine like terms: 0.336935291 + 0.140625 = 0.477560291 x = 0.477560291 Simplifying x = 0.477560291

Subproblem 2

x + -0.140625 = -0.336935291 Simplifying x + -0.140625 = -0.336935291 Reorder the terms: -0.140625 + x = -0.336935291 Solving -0.140625 + x = -0.336935291 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.140625' to each side of the equation. -0.140625 + 0.140625 + x = -0.336935291 + 0.140625 Combine like terms: -0.140625 + 0.140625 = 0.000000 0.000000 + x = -0.336935291 + 0.140625 x = -0.336935291 + 0.140625 Combine like terms: -0.336935291 + 0.140625 = -0.196310291 x = -0.196310291 Simplifying x = -0.196310291

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.477560291, -0.196310291}

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