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Simplifying -5x + 6 = 16x2 + 24x + 9 Reorder the terms: 6 + -5x = 16x2 + 24x + 9 Reorder the terms: 6 + -5x = 9 + 24x + 16x2 Solving 6 + -5x = 9 + 24x + 16x2 Solving for variable 'x'. Reorder the terms: 6 + -9 + -5x + -24x + -16x2 = 9 + 24x + 16x2 + -9 + -24x + -16x2 Combine like terms: 6 + -9 = -3 -3 + -5x + -24x + -16x2 = 9 + 24x + 16x2 + -9 + -24x + -16x2 Combine like terms: -5x + -24x = -29x -3 + -29x + -16x2 = 9 + 24x + 16x2 + -9 + -24x + -16x2 Reorder the terms: -3 + -29x + -16x2 = 9 + -9 + 24x + -24x + 16x2 + -16x2 Combine like terms: 9 + -9 = 0 -3 + -29x + -16x2 = 0 + 24x + -24x + 16x2 + -16x2 -3 + -29x + -16x2 = 24x + -24x + 16x2 + -16x2 Combine like terms: 24x + -24x = 0 -3 + -29x + -16x2 = 0 + 16x2 + -16x2 -3 + -29x + -16x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 -3 + -29x + -16x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(3 + 29x + 16x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(3 + 29x + 16x2)' equal to zero and attempt to solve: Simplifying 3 + 29x + 16x2 = 0 Solving 3 + 29x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.1875 + 1.8125x + x2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + 1.8125x + -0.1875 + x2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + 1.8125x + x2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + 1.8125x + x2 = 0 + -0.1875 1.8125x + x2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 1.8125x + x2 = -0.1875 The x term is 1.8125x. Take half its coefficient (0.90625). Square it (0.8212890625) and add it to both sides. Add '0.8212890625' to each side of the equation. 1.8125x + 0.8212890625 + x2 = -0.1875 + 0.8212890625 Reorder the terms: 0.8212890625 + 1.8125x + x2 = -0.1875 + 0.8212890625 Combine like terms: -0.1875 + 0.8212890625 = 0.6337890625 0.8212890625 + 1.8125x + x2 = 0.6337890625 Factor a perfect square on the left side: (x + 0.90625)(x + 0.90625) = 0.6337890625 Calculate the square root of the right side: 0.7961087 Break this problem into two subproblems by setting (x + 0.90625) equal to 0.7961087 and -0.7961087.Subproblem 1
x + 0.90625 = 0.7961087 Simplifying x + 0.90625 = 0.7961087 Reorder the terms: 0.90625 + x = 0.7961087 Solving 0.90625 + x = 0.7961087 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.90625' to each side of the equation. 0.90625 + -0.90625 + x = 0.7961087 + -0.90625 Combine like terms: 0.90625 + -0.90625 = 0.00000 0.00000 + x = 0.7961087 + -0.90625 x = 0.7961087 + -0.90625 Combine like terms: 0.7961087 + -0.90625 = -0.1101413 x = -0.1101413 Simplifying x = -0.1101413Subproblem 2
x + 0.90625 = -0.7961087 Simplifying x + 0.90625 = -0.7961087 Reorder the terms: 0.90625 + x = -0.7961087 Solving 0.90625 + x = -0.7961087 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.90625' to each side of the equation. 0.90625 + -0.90625 + x = -0.7961087 + -0.90625 Combine like terms: 0.90625 + -0.90625 = 0.00000 0.00000 + x = -0.7961087 + -0.90625 x = -0.7961087 + -0.90625 Combine like terms: -0.7961087 + -0.90625 = -1.7023587 x = -1.7023587 Simplifying x = -1.7023587Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.1101413, -1.7023587}Solution
x = {-0.1101413, -1.7023587}
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