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Simplifying 4x + 7 = (x + 5)(x + 1) Reorder the terms: 7 + 4x = (x + 5)(x + 1) Reorder the terms: 7 + 4x = (5 + x)(x + 1) Reorder the terms: 7 + 4x = (5 + x)(1 + x) Multiply (5 + x) * (1 + x) 7 + 4x = (5(1 + x) + x(1 + x)) 7 + 4x = ((1 * 5 + x * 5) + x(1 + x)) 7 + 4x = ((5 + 5x) + x(1 + x)) 7 + 4x = (5 + 5x + (1 * x + x * x)) 7 + 4x = (5 + 5x + (1x + x2)) Combine like terms: 5x + 1x = 6x 7 + 4x = (5 + 6x + x2) Solving 7 + 4x = 5 + 6x + x2 Solving for variable 'x'. Reorder the terms: 7 + -5 + 4x + -6x + -1x2 = 5 + 6x + x2 + -5 + -6x + -1x2 Combine like terms: 7 + -5 = 2 2 + 4x + -6x + -1x2 = 5 + 6x + x2 + -5 + -6x + -1x2 Combine like terms: 4x + -6x = -2x 2 + -2x + -1x2 = 5 + 6x + x2 + -5 + -6x + -1x2 Reorder the terms: 2 + -2x + -1x2 = 5 + -5 + 6x + -6x + x2 + -1x2 Combine like terms: 5 + -5 = 0 2 + -2x + -1x2 = 0 + 6x + -6x + x2 + -1x2 2 + -2x + -1x2 = 6x + -6x + x2 + -1x2 Combine like terms: 6x + -6x = 0 2 + -2x + -1x2 = 0 + x2 + -1x2 2 + -2x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 2 + -2x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -2 + 2x + x2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 2x + 2 + x2 = 0 + 2 Reorder the terms: -2 + 2 + 2x + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 2x + x2 = 0 + 2 2x + x2 = 0 + 2 Combine like terms: 0 + 2 = 2 2x + x2 = 2 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 2 + 1 Reorder the terms: 1 + 2x + x2 = 2 + 1 Combine like terms: 2 + 1 = 3 1 + 2x + x2 = 3 Factor a perfect square on the left side: (x + 1)(x + 1) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (x + 1) equal to 1.732050808 and -1.732050808.Subproblem 1
x + 1 = 1.732050808 Simplifying x + 1 = 1.732050808 Reorder the terms: 1 + x = 1.732050808 Solving 1 + x = 1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 1.732050808 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.732050808 + -1 x = 1.732050808 + -1 Combine like terms: 1.732050808 + -1 = 0.732050808 x = 0.732050808 Simplifying x = 0.732050808Subproblem 2
x + 1 = -1.732050808 Simplifying x + 1 = -1.732050808 Reorder the terms: 1 + x = -1.732050808 Solving 1 + x = -1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -1.732050808 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.732050808 + -1 x = -1.732050808 + -1 Combine like terms: -1.732050808 + -1 = -2.732050808 x = -2.732050808 Simplifying x = -2.732050808Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.732050808, -2.732050808}
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