32x^2-7x-15=0

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Solution for 32x^2-7x-15=0 equation:


Simplifying
32x2 + -7x + -15 = 0

Reorder the terms:
-15 + -7x + 32x2 = 0

Solving
-15 + -7x + 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'.
-0.46875 + -0.21875x + x2 = 0

Move the constant term to the right:

Add '0.46875' to each side of the equation.
-0.46875 + -0.21875x + 0.46875 + x2 = 0 + 0.46875

Reorder the terms:
-0.46875 + 0.46875 + -0.21875x + x2 = 0 + 0.46875

Combine like terms: -0.46875 + 0.46875 = 0.00000
0.00000 + -0.21875x + x2 = 0 + 0.46875
-0.21875x + x2 = 0 + 0.46875

Combine like terms: 0 + 0.46875 = 0.46875
-0.21875x + x2 = 0.46875

The x term is -0.21875x.  Take half its coefficient (-0.109375).
Square it (0.01196289063) and add it to both sides.

Add '0.01196289063' to each side of the equation.
-0.21875x + 0.01196289063 + x2 = 0.46875 + 0.01196289063

Reorder the terms:
0.01196289063 + -0.21875x + x2 = 0.46875 + 0.01196289063

Combine like terms: 0.46875 + 0.01196289063 = 0.48071289063
0.01196289063 + -0.21875x + x2 = 0.48071289063

Factor a perfect square on the left side:
(x + -0.109375)(x + -0.109375) = 0.48071289063

Calculate the square root of the right side: 0.693334617

Break this problem into two subproblems by setting 
(x + -0.109375) equal to 0.693334617 and -0.693334617.

Subproblem 1

x + -0.109375 = 0.693334617 Simplifying x + -0.109375 = 0.693334617 Reorder the terms: -0.109375 + x = 0.693334617 Solving -0.109375 + x = 0.693334617 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.109375' to each side of the equation. -0.109375 + 0.109375 + x = 0.693334617 + 0.109375 Combine like terms: -0.109375 + 0.109375 = 0.000000 0.000000 + x = 0.693334617 + 0.109375 x = 0.693334617 + 0.109375 Combine like terms: 0.693334617 + 0.109375 = 0.802709617 x = 0.802709617 Simplifying x = 0.802709617

Subproblem 2

x + -0.109375 = -0.693334617 Simplifying x + -0.109375 = -0.693334617 Reorder the terms: -0.109375 + x = -0.693334617 Solving -0.109375 + x = -0.693334617 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.109375' to each side of the equation. -0.109375 + 0.109375 + x = -0.693334617 + 0.109375 Combine like terms: -0.109375 + 0.109375 = 0.000000 0.000000 + x = -0.693334617 + 0.109375 x = -0.693334617 + 0.109375 Combine like terms: -0.693334617 + 0.109375 = -0.583959617 x = -0.583959617 Simplifying x = -0.583959617

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.802709617, -0.583959617}

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