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Simplifying 82 = 10x(x + -1) + 22 + 4x + 14 Reorder the terms: 82 = 10x(-1 + x) + 22 + 4x + 14 82 = (-1 * 10x + x * 10x) + 22 + 4x + 14 82 = (-10x + 10x2) + 22 + 4x + 14 Reorder the terms: 82 = 22 + 14 + -10x + 4x + 10x2 Combine like terms: 22 + 14 = 36 82 = 36 + -10x + 4x + 10x2 Combine like terms: -10x + 4x = -6x 82 = 36 + -6x + 10x2 Solving 82 = 36 + -6x + 10x2 Solving for variable 'x'. Combine like terms: 82 + -36 = 46 46 + 6x + -10x2 = 36 + -6x + 10x2 + -36 + 6x + -10x2 Reorder the terms: 46 + 6x + -10x2 = 36 + -36 + -6x + 6x + 10x2 + -10x2 Combine like terms: 36 + -36 = 0 46 + 6x + -10x2 = 0 + -6x + 6x + 10x2 + -10x2 46 + 6x + -10x2 = -6x + 6x + 10x2 + -10x2 Combine like terms: -6x + 6x = 0 46 + 6x + -10x2 = 0 + 10x2 + -10x2 46 + 6x + -10x2 = 10x2 + -10x2 Combine like terms: 10x2 + -10x2 = 0 46 + 6x + -10x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(23 + 3x + -5x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(23 + 3x + -5x2)' equal to zero and attempt to solve: Simplifying 23 + 3x + -5x2 = 0 Solving 23 + 3x + -5x2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -4.6 + -0.6x + x2 = 0 Move the constant term to the right: Add '4.6' to each side of the equation. -4.6 + -0.6x + 4.6 + x2 = 0 + 4.6 Reorder the terms: -4.6 + 4.6 + -0.6x + x2 = 0 + 4.6 Combine like terms: -4.6 + 4.6 = 0.0 0.0 + -0.6x + x2 = 0 + 4.6 -0.6x + x2 = 0 + 4.6 Combine like terms: 0 + 4.6 = 4.6 -0.6x + x2 = 4.6 The x term is -0.6x. Take half its coefficient (-0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. -0.6x + 0.09 + x2 = 4.6 + 0.09 Reorder the terms: 0.09 + -0.6x + x2 = 4.6 + 0.09 Combine like terms: 4.6 + 0.09 = 4.69 0.09 + -0.6x + x2 = 4.69 Factor a perfect square on the left side: (x + -0.3)(x + -0.3) = 4.69 Calculate the square root of the right side: 2.165640783 Break this problem into two subproblems by setting (x + -0.3) equal to 2.165640783 and -2.165640783.Subproblem 1
x + -0.3 = 2.165640783 Simplifying x + -0.3 = 2.165640783 Reorder the terms: -0.3 + x = 2.165640783 Solving -0.3 + x = 2.165640783 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + x = 2.165640783 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + x = 2.165640783 + 0.3 x = 2.165640783 + 0.3 Combine like terms: 2.165640783 + 0.3 = 2.465640783 x = 2.465640783 Simplifying x = 2.465640783Subproblem 2
x + -0.3 = -2.165640783 Simplifying x + -0.3 = -2.165640783 Reorder the terms: -0.3 + x = -2.165640783 Solving -0.3 + x = -2.165640783 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + x = -2.165640783 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + x = -2.165640783 + 0.3 x = -2.165640783 + 0.3 Combine like terms: -2.165640783 + 0.3 = -1.865640783 x = -1.865640783 Simplifying x = -1.865640783Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.465640783, -1.865640783}Solution
x = {2.465640783, -1.865640783}
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