-0.5*32x^2+12x+22=23.76

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Solution for -0.5*32x^2+12x+22=23.76 equation:


Simplifying
-0.5 * 32x2 + 12x + 22 = 23.76

Multiply -0.5 * 32
-16x2 + 12x + 22 = 23.76

Reorder the terms:
22 + 12x + -16x2 = 23.76

Solving
22 + 12x + -16x2 = 23.76

Solving for variable 'x'.

Reorder the terms:
22 + -23.76 + 12x + -16x2 = 23.76 + -23.76

Combine like terms: 22 + -23.76 = -1.76
-1.76 + 12x + -16x2 = 23.76 + -23.76

Combine like terms: 23.76 + -23.76 = 0.00
-1.76 + 12x + -16x2 = 0.00

Begin completing the square.  Divide all terms by
-16 the coefficient of the squared term: 

Divide each side by '-16'.
0.11 + -0.75x + x2 = 0.00

Move the constant term to the right:

Add '-0.11' to each side of the equation.
0.11 + -0.75x + -0.11 + x2 = 0.00 + -0.11

Reorder the terms:
0.11 + -0.11 + -0.75x + x2 = 0.00 + -0.11

Combine like terms: 0.11 + -0.11 = 0.00
0.00 + -0.75x + x2 = 0.00 + -0.11
-0.75x + x2 = 0.00 + -0.11

Combine like terms: 0.00 + -0.11 = -0.11
-0.75x + x2 = -0.11

The x term is -0.75x.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75x + 0.140625 + x2 = -0.11 + 0.140625

Reorder the terms:
0.140625 + -0.75x + x2 = -0.11 + 0.140625

Combine like terms: -0.11 + 0.140625 = 0.030625
0.140625 + -0.75x + x2 = 0.030625

Factor a perfect square on the left side:
(x + -0.375)(x + -0.375) = 0.030625

Calculate the square root of the right side: 0.175

Break this problem into two subproblems by setting 
(x + -0.375) equal to 0.175 and -0.175.

Subproblem 1

x + -0.375 = 0.175 Simplifying x + -0.375 = 0.175 Reorder the terms: -0.375 + x = 0.175 Solving -0.375 + x = 0.175 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 0.175 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 0.175 + 0.375 x = 0.175 + 0.375 Combine like terms: 0.175 + 0.375 = 0.55 x = 0.55 Simplifying x = 0.55

Subproblem 2

x + -0.375 = -0.175 Simplifying x + -0.375 = -0.175 Reorder the terms: -0.375 + x = -0.175 Solving -0.375 + x = -0.175 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -0.175 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -0.175 + 0.375 x = -0.175 + 0.375 Combine like terms: -0.175 + 0.375 = 0.2 x = 0.2 Simplifying x = 0.2

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.55, 0.2}

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