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Simplifying 24 = (14 + -2x)(10 + -2x) Multiply (14 + -2x) * (10 + -2x) 24 = (14(10 + -2x) + -2x * (10 + -2x)) 24 = ((10 * 14 + -2x * 14) + -2x * (10 + -2x)) 24 = ((140 + -28x) + -2x * (10 + -2x)) 24 = (140 + -28x + (10 * -2x + -2x * -2x)) 24 = (140 + -28x + (-20x + 4x2)) Combine like terms: -28x + -20x = -48x 24 = (140 + -48x + 4x2) Solving 24 = 140 + -48x + 4x2 Solving for variable 'x'. Combine like terms: 24 + -140 = -116 -116 + 48x + -4x2 = 140 + -48x + 4x2 + -140 + 48x + -4x2 Reorder the terms: -116 + 48x + -4x2 = 140 + -140 + -48x + 48x + 4x2 + -4x2 Combine like terms: 140 + -140 = 0 -116 + 48x + -4x2 = 0 + -48x + 48x + 4x2 + -4x2 -116 + 48x + -4x2 = -48x + 48x + 4x2 + -4x2 Combine like terms: -48x + 48x = 0 -116 + 48x + -4x2 = 0 + 4x2 + -4x2 -116 + 48x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 -116 + 48x + -4x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-29 + 12x + -1x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-29 + 12x + -1x2)' equal to zero and attempt to solve: Simplifying -29 + 12x + -1x2 = 0 Solving -29 + 12x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 29 + -12x + x2 = 0 Move the constant term to the right: Add '-29' to each side of the equation. 29 + -12x + -29 + x2 = 0 + -29 Reorder the terms: 29 + -29 + -12x + x2 = 0 + -29 Combine like terms: 29 + -29 = 0 0 + -12x + x2 = 0 + -29 -12x + x2 = 0 + -29 Combine like terms: 0 + -29 = -29 -12x + x2 = -29 The x term is -12x. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12x + 36 + x2 = -29 + 36 Reorder the terms: 36 + -12x + x2 = -29 + 36 Combine like terms: -29 + 36 = 7 36 + -12x + x2 = 7 Factor a perfect square on the left side: (x + -6)(x + -6) = 7 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (x + -6) equal to 2.645751311 and -2.645751311.Subproblem 1
x + -6 = 2.645751311 Simplifying x + -6 = 2.645751311 Reorder the terms: -6 + x = 2.645751311 Solving -6 + x = 2.645751311 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + x = 2.645751311 + 6 Combine like terms: -6 + 6 = 0 0 + x = 2.645751311 + 6 x = 2.645751311 + 6 Combine like terms: 2.645751311 + 6 = 8.645751311 x = 8.645751311 Simplifying x = 8.645751311Subproblem 2
x + -6 = -2.645751311 Simplifying x + -6 = -2.645751311 Reorder the terms: -6 + x = -2.645751311 Solving -6 + x = -2.645751311 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + x = -2.645751311 + 6 Combine like terms: -6 + 6 = 0 0 + x = -2.645751311 + 6 x = -2.645751311 + 6 Combine like terms: -2.645751311 + 6 = 3.354248689 x = 3.354248689 Simplifying x = 3.354248689Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.645751311, 3.354248689}Solution
x = {8.645751311, 3.354248689}
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