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Simplifying 24 = (14 + -1x)(10 + -1x) Multiply (14 + -1x) * (10 + -1x) 24 = (14(10 + -1x) + -1x * (10 + -1x)) 24 = ((10 * 14 + -1x * 14) + -1x * (10 + -1x)) 24 = ((140 + -14x) + -1x * (10 + -1x)) 24 = (140 + -14x + (10 * -1x + -1x * -1x)) 24 = (140 + -14x + (-10x + 1x2)) Combine like terms: -14x + -10x = -24x 24 = (140 + -24x + 1x2) Solving 24 = 140 + -24x + 1x2 Solving for variable 'x'. Combine like terms: 24 + -140 = -116 -116 + 24x + -1x2 = 140 + -24x + 1x2 + -140 + 24x + -1x2 Reorder the terms: -116 + 24x + -1x2 = 140 + -140 + -24x + 24x + 1x2 + -1x2 Combine like terms: 140 + -140 = 0 -116 + 24x + -1x2 = 0 + -24x + 24x + 1x2 + -1x2 -116 + 24x + -1x2 = -24x + 24x + 1x2 + -1x2 Combine like terms: -24x + 24x = 0 -116 + 24x + -1x2 = 0 + 1x2 + -1x2 -116 + 24x + -1x2 = 1x2 + -1x2 Combine like terms: 1x2 + -1x2 = 0 -116 + 24x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 116 + -24x + x2 = 0 Move the constant term to the right: Add '-116' to each side of the equation. 116 + -24x + -116 + x2 = 0 + -116 Reorder the terms: 116 + -116 + -24x + x2 = 0 + -116 Combine like terms: 116 + -116 = 0 0 + -24x + x2 = 0 + -116 -24x + x2 = 0 + -116 Combine like terms: 0 + -116 = -116 -24x + x2 = -116 The x term is -24x. Take half its coefficient (-12). Square it (144) and add it to both sides. Add '144' to each side of the equation. -24x + 144 + x2 = -116 + 144 Reorder the terms: 144 + -24x + x2 = -116 + 144 Combine like terms: -116 + 144 = 28 144 + -24x + x2 = 28 Factor a perfect square on the left side: (x + -12)(x + -12) = 28 Calculate the square root of the right side: 5.291502622 Break this problem into two subproblems by setting (x + -12) equal to 5.291502622 and -5.291502622.Subproblem 1
x + -12 = 5.291502622 Simplifying x + -12 = 5.291502622 Reorder the terms: -12 + x = 5.291502622 Solving -12 + x = 5.291502622 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x = 5.291502622 + 12 Combine like terms: -12 + 12 = 0 0 + x = 5.291502622 + 12 x = 5.291502622 + 12 Combine like terms: 5.291502622 + 12 = 17.291502622 x = 17.291502622 Simplifying x = 17.291502622Subproblem 2
x + -12 = -5.291502622 Simplifying x + -12 = -5.291502622 Reorder the terms: -12 + x = -5.291502622 Solving -12 + x = -5.291502622 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x = -5.291502622 + 12 Combine like terms: -12 + 12 = 0 0 + x = -5.291502622 + 12 x = -5.291502622 + 12 Combine like terms: -5.291502622 + 12 = 6.708497378 x = 6.708497378 Simplifying x = 6.708497378Solution
The solution to the problem is based on the solutions from the subproblems. x = {17.291502622, 6.708497378}
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