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Simplifying 0 = 450 + -180x + 12x2 Solving 0 = 450 + -180x + 12x2 Solving for variable 'x'. Combine like terms: 0 + -450 = -450 -450 + 180x + -12x2 = 450 + -180x + 12x2 + -450 + 180x + -12x2 Reorder the terms: -450 + 180x + -12x2 = 450 + -450 + -180x + 180x + 12x2 + -12x2 Combine like terms: 450 + -450 = 0 -450 + 180x + -12x2 = 0 + -180x + 180x + 12x2 + -12x2 -450 + 180x + -12x2 = -180x + 180x + 12x2 + -12x2 Combine like terms: -180x + 180x = 0 -450 + 180x + -12x2 = 0 + 12x2 + -12x2 -450 + 180x + -12x2 = 12x2 + -12x2 Combine like terms: 12x2 + -12x2 = 0 -450 + 180x + -12x2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-75 + 30x + -2x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-75 + 30x + -2x2)' equal to zero and attempt to solve: Simplifying -75 + 30x + -2x2 = 0 Solving -75 + 30x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 37.5 + -15x + x2 = 0 Move the constant term to the right: Add '-37.5' to each side of the equation. 37.5 + -15x + -37.5 + x2 = 0 + -37.5 Reorder the terms: 37.5 + -37.5 + -15x + x2 = 0 + -37.5 Combine like terms: 37.5 + -37.5 = 0.0 0.0 + -15x + x2 = 0 + -37.5 -15x + x2 = 0 + -37.5 Combine like terms: 0 + -37.5 = -37.5 -15x + x2 = -37.5 The x term is -15x. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15x + 56.25 + x2 = -37.5 + 56.25 Reorder the terms: 56.25 + -15x + x2 = -37.5 + 56.25 Combine like terms: -37.5 + 56.25 = 18.75 56.25 + -15x + x2 = 18.75 Factor a perfect square on the left side: (x + -7.5)(x + -7.5) = 18.75 Calculate the square root of the right side: 4.330127019 Break this problem into two subproblems by setting (x + -7.5) equal to 4.330127019 and -4.330127019.Subproblem 1
x + -7.5 = 4.330127019 Simplifying x + -7.5 = 4.330127019 Reorder the terms: -7.5 + x = 4.330127019 Solving -7.5 + x = 4.330127019 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x = 4.330127019 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = 4.330127019 + 7.5 x = 4.330127019 + 7.5 Combine like terms: 4.330127019 + 7.5 = 11.830127019 x = 11.830127019 Simplifying x = 11.830127019Subproblem 2
x + -7.5 = -4.330127019 Simplifying x + -7.5 = -4.330127019 Reorder the terms: -7.5 + x = -4.330127019 Solving -7.5 + x = -4.330127019 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + x = -4.330127019 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + x = -4.330127019 + 7.5 x = -4.330127019 + 7.5 Combine like terms: -4.330127019 + 7.5 = 3.169872981 x = 3.169872981 Simplifying x = 3.169872981Solution
The solution to the problem is based on the solutions from the subproblems. x = {11.830127019, 3.169872981}Solution
x = {11.830127019, 3.169872981}
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