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Simplifying (2x + 5)(x + 1) = 52 Reorder the terms: (5 + 2x)(x + 1) = 52 Reorder the terms: (5 + 2x)(1 + x) = 52 Multiply (5 + 2x) * (1 + x) (5(1 + x) + 2x * (1 + x)) = 52 ((1 * 5 + x * 5) + 2x * (1 + x)) = 52 ((5 + 5x) + 2x * (1 + x)) = 52 (5 + 5x + (1 * 2x + x * 2x)) = 52 (5 + 5x + (2x + 2x2)) = 52 Combine like terms: 5x + 2x = 7x (5 + 7x + 2x2) = 52 Solving 5 + 7x + 2x2 = 52 Solving for variable 'x'. Reorder the terms: 5 + -52 + 7x + 2x2 = 52 + -52 Combine like terms: 5 + -52 = -47 -47 + 7x + 2x2 = 52 + -52 Combine like terms: 52 + -52 = 0 -47 + 7x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -23.5 + 3.5x + x2 = 0 Move the constant term to the right: Add '23.5' to each side of the equation. -23.5 + 3.5x + 23.5 + x2 = 0 + 23.5 Reorder the terms: -23.5 + 23.5 + 3.5x + x2 = 0 + 23.5 Combine like terms: -23.5 + 23.5 = 0.0 0.0 + 3.5x + x2 = 0 + 23.5 3.5x + x2 = 0 + 23.5 Combine like terms: 0 + 23.5 = 23.5 3.5x + x2 = 23.5 The x term is 3.5x. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5x + 3.0625 + x2 = 23.5 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = 23.5 + 3.0625 Combine like terms: 23.5 + 3.0625 = 26.5625 3.0625 + 3.5x + x2 = 26.5625 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 26.5625 Calculate the square root of the right side: 5.153882032 Break this problem into two subproblems by setting (x + 1.75) equal to 5.153882032 and -5.153882032.Subproblem 1
x + 1.75 = 5.153882032 Simplifying x + 1.75 = 5.153882032 Reorder the terms: 1.75 + x = 5.153882032 Solving 1.75 + x = 5.153882032 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 5.153882032 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 5.153882032 + -1.75 x = 5.153882032 + -1.75 Combine like terms: 5.153882032 + -1.75 = 3.403882032 x = 3.403882032 Simplifying x = 3.403882032Subproblem 2
x + 1.75 = -5.153882032 Simplifying x + 1.75 = -5.153882032 Reorder the terms: 1.75 + x = -5.153882032 Solving 1.75 + x = -5.153882032 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -5.153882032 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -5.153882032 + -1.75 x = -5.153882032 + -1.75 Combine like terms: -5.153882032 + -1.75 = -6.903882032 x = -6.903882032 Simplifying x = -6.903882032Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.403882032, -6.903882032}
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