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Simplifying 6x2 + -18x + -24 = 75 Reorder the terms: -24 + -18x + 6x2 = 75 Solving -24 + -18x + 6x2 = 75 Solving for variable 'x'. Reorder the terms: -24 + -75 + -18x + 6x2 = 75 + -75 Combine like terms: -24 + -75 = -99 -99 + -18x + 6x2 = 75 + -75 Combine like terms: 75 + -75 = 0 -99 + -18x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-33 + -6x + 2x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-33 + -6x + 2x2)' equal to zero and attempt to solve: Simplifying -33 + -6x + 2x2 = 0 Solving -33 + -6x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -16.5 + -3x + x2 = 0 Move the constant term to the right: Add '16.5' to each side of the equation. -16.5 + -3x + 16.5 + x2 = 0 + 16.5 Reorder the terms: -16.5 + 16.5 + -3x + x2 = 0 + 16.5 Combine like terms: -16.5 + 16.5 = 0.0 0.0 + -3x + x2 = 0 + 16.5 -3x + x2 = 0 + 16.5 Combine like terms: 0 + 16.5 = 16.5 -3x + x2 = 16.5 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = 16.5 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 16.5 + 2.25 Combine like terms: 16.5 + 2.25 = 18.75 2.25 + -3x + x2 = 18.75 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 18.75 Calculate the square root of the right side: 4.330127019 Break this problem into two subproblems by setting (x + -1.5) equal to 4.330127019 and -4.330127019.Subproblem 1
x + -1.5 = 4.330127019 Simplifying x + -1.5 = 4.330127019 Reorder the terms: -1.5 + x = 4.330127019 Solving -1.5 + x = 4.330127019 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 4.330127019 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 4.330127019 + 1.5 x = 4.330127019 + 1.5 Combine like terms: 4.330127019 + 1.5 = 5.830127019 x = 5.830127019 Simplifying x = 5.830127019Subproblem 2
x + -1.5 = -4.330127019 Simplifying x + -1.5 = -4.330127019 Reorder the terms: -1.5 + x = -4.330127019 Solving -1.5 + x = -4.330127019 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -4.330127019 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -4.330127019 + 1.5 x = -4.330127019 + 1.5 Combine like terms: -4.330127019 + 1.5 = -2.830127019 x = -2.830127019 Simplifying x = -2.830127019Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.830127019, -2.830127019}Solution
x = {5.830127019, -2.830127019}
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