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Simplifying 16x2 = 20x2 + 24x + 5 Reorder the terms: 16x2 = 5 + 24x + 20x2 Solving 16x2 = 5 + 24x + 20x2 Solving for variable 'x'. Reorder the terms: -5 + -24x + 16x2 + -20x2 = 5 + 24x + 20x2 + -5 + -24x + -20x2 Combine like terms: 16x2 + -20x2 = -4x2 -5 + -24x + -4x2 = 5 + 24x + 20x2 + -5 + -24x + -20x2 Reorder the terms: -5 + -24x + -4x2 = 5 + -5 + 24x + -24x + 20x2 + -20x2 Combine like terms: 5 + -5 = 0 -5 + -24x + -4x2 = 0 + 24x + -24x + 20x2 + -20x2 -5 + -24x + -4x2 = 24x + -24x + 20x2 + -20x2 Combine like terms: 24x + -24x = 0 -5 + -24x + -4x2 = 0 + 20x2 + -20x2 -5 + -24x + -4x2 = 20x2 + -20x2 Combine like terms: 20x2 + -20x2 = 0 -5 + -24x + -4x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(5 + 24x + 4x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(5 + 24x + 4x2)' equal to zero and attempt to solve: Simplifying 5 + 24x + 4x2 = 0 Solving 5 + 24x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.25 + 6x + x2 = 0 Move the constant term to the right: Add '-1.25' to each side of the equation. 1.25 + 6x + -1.25 + x2 = 0 + -1.25 Reorder the terms: 1.25 + -1.25 + 6x + x2 = 0 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + 6x + x2 = 0 + -1.25 6x + x2 = 0 + -1.25 Combine like terms: 0 + -1.25 = -1.25 6x + x2 = -1.25 The x term is 6x. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6x + 9 + x2 = -1.25 + 9 Reorder the terms: 9 + 6x + x2 = -1.25 + 9 Combine like terms: -1.25 + 9 = 7.75 9 + 6x + x2 = 7.75 Factor a perfect square on the left side: (x + 3)(x + 3) = 7.75 Calculate the square root of the right side: 2.783882181 Break this problem into two subproblems by setting (x + 3) equal to 2.783882181 and -2.783882181.Subproblem 1
x + 3 = 2.783882181 Simplifying x + 3 = 2.783882181 Reorder the terms: 3 + x = 2.783882181 Solving 3 + x = 2.783882181 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = 2.783882181 + -3 Combine like terms: 3 + -3 = 0 0 + x = 2.783882181 + -3 x = 2.783882181 + -3 Combine like terms: 2.783882181 + -3 = -0.216117819 x = -0.216117819 Simplifying x = -0.216117819Subproblem 2
x + 3 = -2.783882181 Simplifying x + 3 = -2.783882181 Reorder the terms: 3 + x = -2.783882181 Solving 3 + x = -2.783882181 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = -2.783882181 + -3 Combine like terms: 3 + -3 = 0 0 + x = -2.783882181 + -3 x = -2.783882181 + -3 Combine like terms: -2.783882181 + -3 = -5.783882181 x = -5.783882181 Simplifying x = -5.783882181Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.216117819, -5.783882181}Solution
x = {-0.216117819, -5.783882181}
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