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Simplifying 3x + 12 = 24x(2 + x) Reorder the terms: 12 + 3x = 24x(2 + x) 12 + 3x = (2 * 24x + x * 24x) 12 + 3x = (48x + 24x2) Solving 12 + 3x = 48x + 24x2 Solving for variable 'x'. Combine like terms: 3x + -48x = -45x 12 + -45x + -24x2 = 48x + 24x2 + -48x + -24x2 Reorder the terms: 12 + -45x + -24x2 = 48x + -48x + 24x2 + -24x2 Combine like terms: 48x + -48x = 0 12 + -45x + -24x2 = 0 + 24x2 + -24x2 12 + -45x + -24x2 = 24x2 + -24x2 Combine like terms: 24x2 + -24x2 = 0 12 + -45x + -24x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(4 + -15x + -8x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(4 + -15x + -8x2)' equal to zero and attempt to solve: Simplifying 4 + -15x + -8x2 = 0 Solving 4 + -15x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.5 + 1.875x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 1.875x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 1.875x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 1.875x + x2 = 0 + 0.5 1.875x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 1.875x + x2 = 0.5 The x term is 1.875x. Take half its coefficient (0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. 1.875x + 0.87890625 + x2 = 0.5 + 0.87890625 Reorder the terms: 0.87890625 + 1.875x + x2 = 0.5 + 0.87890625 Combine like terms: 0.5 + 0.87890625 = 1.37890625 0.87890625 + 1.875x + x2 = 1.37890625 Factor a perfect square on the left side: (x + 0.9375)(x + 0.9375) = 1.37890625 Calculate the square root of the right side: 1.174268389 Break this problem into two subproblems by setting (x + 0.9375) equal to 1.174268389 and -1.174268389.Subproblem 1
x + 0.9375 = 1.174268389 Simplifying x + 0.9375 = 1.174268389 Reorder the terms: 0.9375 + x = 1.174268389 Solving 0.9375 + x = 1.174268389 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + x = 1.174268389 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + x = 1.174268389 + -0.9375 x = 1.174268389 + -0.9375 Combine like terms: 1.174268389 + -0.9375 = 0.236768389 x = 0.236768389 Simplifying x = 0.236768389Subproblem 2
x + 0.9375 = -1.174268389 Simplifying x + 0.9375 = -1.174268389 Reorder the terms: 0.9375 + x = -1.174268389 Solving 0.9375 + x = -1.174268389 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + x = -1.174268389 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + x = -1.174268389 + -0.9375 x = -1.174268389 + -0.9375 Combine like terms: -1.174268389 + -0.9375 = -2.111768389 x = -2.111768389 Simplifying x = -2.111768389Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.236768389, -2.111768389}Solution
x = {0.236768389, -2.111768389}
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