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Simplifying (2X + 3)(5X + 4) = 20 Reorder the terms: (3 + 2X)(5X + 4) = 20 Reorder the terms: (3 + 2X)(4 + 5X) = 20 Multiply (3 + 2X) * (4 + 5X) (3(4 + 5X) + 2X * (4 + 5X)) = 20 ((4 * 3 + 5X * 3) + 2X * (4 + 5X)) = 20 ((12 + 15X) + 2X * (4 + 5X)) = 20 (12 + 15X + (4 * 2X + 5X * 2X)) = 20 (12 + 15X + (8X + 10X2)) = 20 Combine like terms: 15X + 8X = 23X (12 + 23X + 10X2) = 20 Solving 12 + 23X + 10X2 = 20 Solving for variable 'X'. Reorder the terms: 12 + -20 + 23X + 10X2 = 20 + -20 Combine like terms: 12 + -20 = -8 -8 + 23X + 10X2 = 20 + -20 Combine like terms: 20 + -20 = 0 -8 + 23X + 10X2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.8 + 2.3X + X2 = 0.0 Move the constant term to the right: Add '0.8' to each side of the equation. -0.8 + 2.3X + 0.8 + X2 = 0.0 + 0.8 Reorder the terms: -0.8 + 0.8 + 2.3X + X2 = 0.0 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + 2.3X + X2 = 0.0 + 0.8 2.3X + X2 = 0.0 + 0.8 Combine like terms: 0.0 + 0.8 = 0.8 2.3X + X2 = 0.8 The X term is 2.3X. Take half its coefficient (1.15). Square it (1.3225) and add it to both sides. Add '1.3225' to each side of the equation. 2.3X + 1.3225 + X2 = 0.8 + 1.3225 Reorder the terms: 1.3225 + 2.3X + X2 = 0.8 + 1.3225 Combine like terms: 0.8 + 1.3225 = 2.1225 1.3225 + 2.3X + X2 = 2.1225 Factor a perfect square on the left side: (X + 1.15)(X + 1.15) = 2.1225 Calculate the square root of the right side: 1.456880228 Break this problem into two subproblems by setting (X + 1.15) equal to 1.456880228 and -1.456880228.Subproblem 1
X + 1.15 = 1.456880228 Simplifying X + 1.15 = 1.456880228 Reorder the terms: 1.15 + X = 1.456880228 Solving 1.15 + X = 1.456880228 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-1.15' to each side of the equation. 1.15 + -1.15 + X = 1.456880228 + -1.15 Combine like terms: 1.15 + -1.15 = 0.00 0.00 + X = 1.456880228 + -1.15 X = 1.456880228 + -1.15 Combine like terms: 1.456880228 + -1.15 = 0.306880228 X = 0.306880228 Simplifying X = 0.306880228Subproblem 2
X + 1.15 = -1.456880228 Simplifying X + 1.15 = -1.456880228 Reorder the terms: 1.15 + X = -1.456880228 Solving 1.15 + X = -1.456880228 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-1.15' to each side of the equation. 1.15 + -1.15 + X = -1.456880228 + -1.15 Combine like terms: 1.15 + -1.15 = 0.00 0.00 + X = -1.456880228 + -1.15 X = -1.456880228 + -1.15 Combine like terms: -1.456880228 + -1.15 = -2.606880228 X = -2.606880228 Simplifying X = -2.606880228Solution
The solution to the problem is based on the solutions from the subproblems. X = {0.306880228, -2.606880228}
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