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Simplifying (5x + 3)(4x + 1) = 103 Reorder the terms: (3 + 5x)(4x + 1) = 103 Reorder the terms: (3 + 5x)(1 + 4x) = 103 Multiply (3 + 5x) * (1 + 4x) (3(1 + 4x) + 5x * (1 + 4x)) = 103 ((1 * 3 + 4x * 3) + 5x * (1 + 4x)) = 103 ((3 + 12x) + 5x * (1 + 4x)) = 103 (3 + 12x + (1 * 5x + 4x * 5x)) = 103 (3 + 12x + (5x + 20x2)) = 103 Combine like terms: 12x + 5x = 17x (3 + 17x + 20x2) = 103 Solving 3 + 17x + 20x2 = 103 Solving for variable 'x'. Reorder the terms: 3 + -103 + 17x + 20x2 = 103 + -103 Combine like terms: 3 + -103 = -100 -100 + 17x + 20x2 = 103 + -103 Combine like terms: 103 + -103 = 0 -100 + 17x + 20x2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -5 + 0.85x + x2 = 0.0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 0.85x + 5 + x2 = 0.0 + 5 Reorder the terms: -5 + 5 + 0.85x + x2 = 0.0 + 5 Combine like terms: -5 + 5 = 0 0 + 0.85x + x2 = 0.0 + 5 0.85x + x2 = 0.0 + 5 Combine like terms: 0.0 + 5 = 5 0.85x + x2 = 5 The x term is 0.85x. Take half its coefficient (0.425). Square it (0.180625) and add it to both sides. Add '0.180625' to each side of the equation. 0.85x + 0.180625 + x2 = 5 + 0.180625 Reorder the terms: 0.180625 + 0.85x + x2 = 5 + 0.180625 Combine like terms: 5 + 0.180625 = 5.180625 0.180625 + 0.85x + x2 = 5.180625 Factor a perfect square on the left side: (x + 0.425)(x + 0.425) = 5.180625 Calculate the square root of the right side: 2.276098636 Break this problem into two subproblems by setting (x + 0.425) equal to 2.276098636 and -2.276098636.Subproblem 1
x + 0.425 = 2.276098636 Simplifying x + 0.425 = 2.276098636 Reorder the terms: 0.425 + x = 2.276098636 Solving 0.425 + x = 2.276098636 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.425' to each side of the equation. 0.425 + -0.425 + x = 2.276098636 + -0.425 Combine like terms: 0.425 + -0.425 = 0.000 0.000 + x = 2.276098636 + -0.425 x = 2.276098636 + -0.425 Combine like terms: 2.276098636 + -0.425 = 1.851098636 x = 1.851098636 Simplifying x = 1.851098636Subproblem 2
x + 0.425 = -2.276098636 Simplifying x + 0.425 = -2.276098636 Reorder the terms: 0.425 + x = -2.276098636 Solving 0.425 + x = -2.276098636 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.425' to each side of the equation. 0.425 + -0.425 + x = -2.276098636 + -0.425 Combine like terms: 0.425 + -0.425 = 0.000 0.000 + x = -2.276098636 + -0.425 x = -2.276098636 + -0.425 Combine like terms: -2.276098636 + -0.425 = -2.701098636 x = -2.701098636 Simplifying x = -2.701098636Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.851098636, -2.701098636}
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