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Simplifying 50x2 * 41 = 2x4 + 5000 Reorder the terms for easier multiplication: 50 * 41x2 = 2x4 + 5000 Multiply 50 * 41 2050x2 = 2x4 + 5000 Reorder the terms: 2050x2 = 5000 + 2x4 Solving 2050x2 = 5000 + 2x4 Solving for variable 'x'. Reorder the terms: -5000 + 2050x2 + -2x4 = 5000 + 2x4 + -5000 + -2x4 Reorder the terms: -5000 + 2050x2 + -2x4 = 5000 + -5000 + 2x4 + -2x4 Combine like terms: 5000 + -5000 = 0 -5000 + 2050x2 + -2x4 = 0 + 2x4 + -2x4 -5000 + 2050x2 + -2x4 = 2x4 + -2x4 Combine like terms: 2x4 + -2x4 = 0 -5000 + 2050x2 + -2x4 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-2500 + 1025x2 + -1x4) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-2500 + 1025x2 + -1x4)' equal to zero and attempt to solve: Simplifying -2500 + 1025x2 + -1x4 = 0 Solving -2500 + 1025x2 + -1x4 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 2500 + -1025x2 + x4 = 0 Move the constant term to the right: Add '-2500' to each side of the equation. 2500 + -1025x2 + -2500 + x4 = 0 + -2500 Reorder the terms: 2500 + -2500 + -1025x2 + x4 = 0 + -2500 Combine like terms: 2500 + -2500 = 0 0 + -1025x2 + x4 = 0 + -2500 -1025x2 + x4 = 0 + -2500 Combine like terms: 0 + -2500 = -2500 -1025x2 + x4 = -2500 The x term is -1025x2. Take half its coefficient (-512.5). Square it (262656.25) and add it to both sides. Add '262656.25' to each side of the equation. -1025x2 + 262656.25 + x4 = -2500 + 262656.25 Reorder the terms: 262656.25 + -1025x2 + x4 = -2500 + 262656.25 Combine like terms: -2500 + 262656.25 = 260156.25 262656.25 + -1025x2 + x4 = 260156.25 Factor a perfect square on the left side: (x2 + -512.5)(x2 + -512.5) = 260156.25 Calculate the square root of the right side: 510.055144078 Break this problem into two subproblems by setting (x2 + -512.5) equal to 510.055144078 and -510.055144078.Subproblem 1
x2 + -512.5 = 510.055144078 Simplifying x2 + -512.5 = 510.055144078 Reorder the terms: -512.5 + x2 = 510.055144078 Solving -512.5 + x2 = 510.055144078 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '512.5' to each side of the equation. -512.5 + 512.5 + x2 = 510.055144078 + 512.5 Combine like terms: -512.5 + 512.5 = 0.0 0.0 + x2 = 510.055144078 + 512.5 x2 = 510.055144078 + 512.5 Combine like terms: 510.055144078 + 512.5 = 1022.555144078 x2 = 1022.555144078 Simplifying x2 = 1022.555144078 Take the square root of each side: x = {-31.977416157, 31.977416157}Subproblem 2
x2 + -512.5 = -510.055144078 Simplifying x2 + -512.5 = -510.055144078 Reorder the terms: -512.5 + x2 = -510.055144078 Solving -512.5 + x2 = -510.055144078 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '512.5' to each side of the equation. -512.5 + 512.5 + x2 = -510.055144078 + 512.5 Combine like terms: -512.5 + 512.5 = 0.0 0.0 + x2 = -510.055144078 + 512.5 x2 = -510.055144078 + 512.5 Combine like terms: -510.055144078 + 512.5 = 2.444855922 x2 = 2.444855922 Simplifying x2 = 2.444855922 Take the square root of each side: x = {-1.563603505, 1.563603505}Solution
The solution to the problem is based on the solutions from the subproblems. x = {-31.977416157, 31.977416157, -1.563603505, 1.563603505}Solution
x = {-31.977416157, 31.977416157, -1.563603505, 1.563603505}
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