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Simplifying x2 + -21x + 109 = 0 Reorder the terms: 109 + -21x + x2 = 0 Solving 109 + -21x + x2 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '-109' to each side of the equation. 109 + -21x + -109 + x2 = 0 + -109 Reorder the terms: 109 + -109 + -21x + x2 = 0 + -109 Combine like terms: 109 + -109 = 0 0 + -21x + x2 = 0 + -109 -21x + x2 = 0 + -109 Combine like terms: 0 + -109 = -109 -21x + x2 = -109 The x term is -21x. Take half its coefficient (-10.5). Square it (110.25) and add it to both sides. Add '110.25' to each side of the equation. -21x + 110.25 + x2 = -109 + 110.25 Reorder the terms: 110.25 + -21x + x2 = -109 + 110.25 Combine like terms: -109 + 110.25 = 1.25 110.25 + -21x + x2 = 1.25 Factor a perfect square on the left side: (x + -10.5)(x + -10.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (x + -10.5) equal to 1.118033989 and -1.118033989.Subproblem 1
x + -10.5 = 1.118033989 Simplifying x + -10.5 = 1.118033989 Reorder the terms: -10.5 + x = 1.118033989 Solving -10.5 + x = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x = 1.118033989 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = 1.118033989 + 10.5 x = 1.118033989 + 10.5 Combine like terms: 1.118033989 + 10.5 = 11.618033989 x = 11.618033989 Simplifying x = 11.618033989Subproblem 2
x + -10.5 = -1.118033989 Simplifying x + -10.5 = -1.118033989 Reorder the terms: -10.5 + x = -1.118033989 Solving -10.5 + x = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x = -1.118033989 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = -1.118033989 + 10.5 x = -1.118033989 + 10.5 Combine like terms: -1.118033989 + 10.5 = 9.381966011 x = 9.381966011 Simplifying x = 9.381966011Solution
The solution to the problem is based on the solutions from the subproblems. x = {11.618033989, 9.381966011}
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