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Simplifying (4x + 1)(2x + 5) = 7x Reorder the terms: (1 + 4x)(2x + 5) = 7x Reorder the terms: (1 + 4x)(5 + 2x) = 7x Multiply (1 + 4x) * (5 + 2x) (1(5 + 2x) + 4x * (5 + 2x)) = 7x ((5 * 1 + 2x * 1) + 4x * (5 + 2x)) = 7x ((5 + 2x) + 4x * (5 + 2x)) = 7x (5 + 2x + (5 * 4x + 2x * 4x)) = 7x (5 + 2x + (20x + 8x2)) = 7x Combine like terms: 2x + 20x = 22x (5 + 22x + 8x2) = 7x Solving 5 + 22x + 8x2 = 7x Solving for variable 'x'. Reorder the terms: 5 + 22x + -7x + 8x2 = 7x + -7x Combine like terms: 22x + -7x = 15x 5 + 15x + 8x2 = 7x + -7x Combine like terms: 7x + -7x = 0 5 + 15x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.625 + 1.875x + x2 = 0 Move the constant term to the right: Add '-0.625' to each side of the equation. 0.625 + 1.875x + -0.625 + x2 = 0 + -0.625 Reorder the terms: 0.625 + -0.625 + 1.875x + x2 = 0 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + 1.875x + x2 = 0 + -0.625 1.875x + x2 = 0 + -0.625 Combine like terms: 0 + -0.625 = -0.625 1.875x + x2 = -0.625 The x term is 1.875x. Take half its coefficient (0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. 1.875x + 0.87890625 + x2 = -0.625 + 0.87890625 Reorder the terms: 0.87890625 + 1.875x + x2 = -0.625 + 0.87890625 Combine like terms: -0.625 + 0.87890625 = 0.25390625 0.87890625 + 1.875x + x2 = 0.25390625 Factor a perfect square on the left side: (x + 0.9375)(x + 0.9375) = 0.25390625 Calculate the square root of the right side: 0.503891109 Break this problem into two subproblems by setting (x + 0.9375) equal to 0.503891109 and -0.503891109.Subproblem 1
x + 0.9375 = 0.503891109 Simplifying x + 0.9375 = 0.503891109 Reorder the terms: 0.9375 + x = 0.503891109 Solving 0.9375 + x = 0.503891109 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + x = 0.503891109 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + x = 0.503891109 + -0.9375 x = 0.503891109 + -0.9375 Combine like terms: 0.503891109 + -0.9375 = -0.433608891 x = -0.433608891 Simplifying x = -0.433608891Subproblem 2
x + 0.9375 = -0.503891109 Simplifying x + 0.9375 = -0.503891109 Reorder the terms: 0.9375 + x = -0.503891109 Solving 0.9375 + x = -0.503891109 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + x = -0.503891109 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + x = -0.503891109 + -0.9375 x = -0.503891109 + -0.9375 Combine like terms: -0.503891109 + -0.9375 = -1.441391109 x = -1.441391109 Simplifying x = -1.441391109Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.433608891, -1.441391109}
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