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Simplifying 4x2 + -40x + 106 = 38 Reorder the terms: 106 + -40x + 4x2 = 38 Solving 106 + -40x + 4x2 = 38 Solving for variable 'x'. Reorder the terms: 106 + -38 + -40x + 4x2 = 38 + -38 Combine like terms: 106 + -38 = 68 68 + -40x + 4x2 = 38 + -38 Combine like terms: 38 + -38 = 0 68 + -40x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(17 + -10x + x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(17 + -10x + x2)' equal to zero and attempt to solve: Simplifying 17 + -10x + x2 = 0 Solving 17 + -10x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-17' to each side of the equation. 17 + -10x + -17 + x2 = 0 + -17 Reorder the terms: 17 + -17 + -10x + x2 = 0 + -17 Combine like terms: 17 + -17 = 0 0 + -10x + x2 = 0 + -17 -10x + x2 = 0 + -17 Combine like terms: 0 + -17 = -17 -10x + x2 = -17 The x term is -10x. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10x + 25 + x2 = -17 + 25 Reorder the terms: 25 + -10x + x2 = -17 + 25 Combine like terms: -17 + 25 = 8 25 + -10x + x2 = 8 Factor a perfect square on the left side: (x + -5)(x + -5) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (x + -5) equal to 2.828427125 and -2.828427125.Subproblem 1
x + -5 = 2.828427125 Simplifying x + -5 = 2.828427125 Reorder the terms: -5 + x = 2.828427125 Solving -5 + x = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 2.828427125 + 5 Combine like terms: -5 + 5 = 0 0 + x = 2.828427125 + 5 x = 2.828427125 + 5 Combine like terms: 2.828427125 + 5 = 7.828427125 x = 7.828427125 Simplifying x = 7.828427125Subproblem 2
x + -5 = -2.828427125 Simplifying x + -5 = -2.828427125 Reorder the terms: -5 + x = -2.828427125 Solving -5 + x = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -2.828427125 + 5 Combine like terms: -5 + 5 = 0 0 + x = -2.828427125 + 5 x = -2.828427125 + 5 Combine like terms: -2.828427125 + 5 = 2.171572875 x = 2.171572875 Simplifying x = 2.171572875Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.828427125, 2.171572875}Solution
x = {7.828427125, 2.171572875}
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