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Simplifying x2 + 30x + 100 = 70x + -1x2 Reorder the terms: 100 + 30x + x2 = 70x + -1x2 Solving 100 + 30x + x2 = 70x + -1x2 Solving for variable 'x'. Reorder the terms: 100 + 30x + -70x + x2 + x2 = 70x + -1x2 + -70x + x2 Combine like terms: 30x + -70x = -40x 100 + -40x + x2 + x2 = 70x + -1x2 + -70x + x2 Combine like terms: x2 + x2 = 2x2 100 + -40x + 2x2 = 70x + -1x2 + -70x + x2 Reorder the terms: 100 + -40x + 2x2 = 70x + -70x + -1x2 + x2 Combine like terms: 70x + -70x = 0 100 + -40x + 2x2 = 0 + -1x2 + x2 100 + -40x + 2x2 = -1x2 + x2 Combine like terms: -1x2 + x2 = 0 100 + -40x + 2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(50 + -20x + x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(50 + -20x + x2)' equal to zero and attempt to solve: Simplifying 50 + -20x + x2 = 0 Solving 50 + -20x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-50' to each side of the equation. 50 + -20x + -50 + x2 = 0 + -50 Reorder the terms: 50 + -50 + -20x + x2 = 0 + -50 Combine like terms: 50 + -50 = 0 0 + -20x + x2 = 0 + -50 -20x + x2 = 0 + -50 Combine like terms: 0 + -50 = -50 -20x + x2 = -50 The x term is -20x. Take half its coefficient (-10). Square it (100) and add it to both sides. Add '100' to each side of the equation. -20x + 100 + x2 = -50 + 100 Reorder the terms: 100 + -20x + x2 = -50 + 100 Combine like terms: -50 + 100 = 50 100 + -20x + x2 = 50 Factor a perfect square on the left side: (x + -10)(x + -10) = 50 Calculate the square root of the right side: 7.071067812 Break this problem into two subproblems by setting (x + -10) equal to 7.071067812 and -7.071067812.Subproblem 1
x + -10 = 7.071067812 Simplifying x + -10 = 7.071067812 Reorder the terms: -10 + x = 7.071067812 Solving -10 + x = 7.071067812 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + x = 7.071067812 + 10 Combine like terms: -10 + 10 = 0 0 + x = 7.071067812 + 10 x = 7.071067812 + 10 Combine like terms: 7.071067812 + 10 = 17.071067812 x = 17.071067812 Simplifying x = 17.071067812Subproblem 2
x + -10 = -7.071067812 Simplifying x + -10 = -7.071067812 Reorder the terms: -10 + x = -7.071067812 Solving -10 + x = -7.071067812 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + x = -7.071067812 + 10 Combine like terms: -10 + 10 = 0 0 + x = -7.071067812 + 10 x = -7.071067812 + 10 Combine like terms: -7.071067812 + 10 = 2.928932188 x = 2.928932188 Simplifying x = 2.928932188Solution
The solution to the problem is based on the solutions from the subproblems. x = {17.071067812, 2.928932188}Solution
x = {17.071067812, 2.928932188}
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