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Simplifying 0 = 3X2 + -24X + 32 Reorder the terms: 0 = 32 + -24X + 3X2 Solving 0 = 32 + -24X + 3X2 Solving for variable 'X'. Combine like terms: 0 + -32 = -32 -32 + 24X + -3X2 = 32 + -24X + 3X2 + -32 + 24X + -3X2 Reorder the terms: -32 + 24X + -3X2 = 32 + -32 + -24X + 24X + 3X2 + -3X2 Combine like terms: 32 + -32 = 0 -32 + 24X + -3X2 = 0 + -24X + 24X + 3X2 + -3X2 -32 + 24X + -3X2 = -24X + 24X + 3X2 + -3X2 Combine like terms: -24X + 24X = 0 -32 + 24X + -3X2 = 0 + 3X2 + -3X2 -32 + 24X + -3X2 = 3X2 + -3X2 Combine like terms: 3X2 + -3X2 = 0 -32 + 24X + -3X2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 10.66666667 + -8X + X2 = 0 Move the constant term to the right: Add '-10.66666667' to each side of the equation. 10.66666667 + -8X + -10.66666667 + X2 = 0 + -10.66666667 Reorder the terms: 10.66666667 + -10.66666667 + -8X + X2 = 0 + -10.66666667 Combine like terms: 10.66666667 + -10.66666667 = 0.00000000 0.00000000 + -8X + X2 = 0 + -10.66666667 -8X + X2 = 0 + -10.66666667 Combine like terms: 0 + -10.66666667 = -10.66666667 -8X + X2 = -10.66666667 The X term is -8X. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8X + 16 + X2 = -10.66666667 + 16 Reorder the terms: 16 + -8X + X2 = -10.66666667 + 16 Combine like terms: -10.66666667 + 16 = 5.33333333 16 + -8X + X2 = 5.33333333 Factor a perfect square on the left side: (X + -4)(X + -4) = 5.33333333 Calculate the square root of the right side: 2.309401076 Break this problem into two subproblems by setting (X + -4) equal to 2.309401076 and -2.309401076.Subproblem 1
X + -4 = 2.309401076 Simplifying X + -4 = 2.309401076 Reorder the terms: -4 + X = 2.309401076 Solving -4 + X = 2.309401076 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + X = 2.309401076 + 4 Combine like terms: -4 + 4 = 0 0 + X = 2.309401076 + 4 X = 2.309401076 + 4 Combine like terms: 2.309401076 + 4 = 6.309401076 X = 6.309401076 Simplifying X = 6.309401076Subproblem 2
X + -4 = -2.309401076 Simplifying X + -4 = -2.309401076 Reorder the terms: -4 + X = -2.309401076 Solving -4 + X = -2.309401076 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + X = -2.309401076 + 4 Combine like terms: -4 + 4 = 0 0 + X = -2.309401076 + 4 X = -2.309401076 + 4 Combine like terms: -2.309401076 + 4 = 1.690598924 X = 1.690598924 Simplifying X = 1.690598924Solution
The solution to the problem is based on the solutions from the subproblems. X = {6.309401076, 1.690598924}
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