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Simplifying (x + 1.5)(x + 0.5) = 20 Reorder the terms: (1.5 + x)(x + 0.5) = 20 Reorder the terms: (1.5 + x)(0.5 + x) = 20 Multiply (1.5 + x) * (0.5 + x) (1.5(0.5 + x) + x(0.5 + x)) = 20 ((0.5 * 1.5 + x * 1.5) + x(0.5 + x)) = 20 ((0.75 + 1.5x) + x(0.5 + x)) = 20 (0.75 + 1.5x + (0.5 * x + x * x)) = 20 (0.75 + 1.5x + (0.5x + x2)) = 20 Combine like terms: 1.5x + 0.5x = 2x (0.75 + 2x + x2) = 20 Solving 0.75 + 2x + x2 = 20 Solving for variable 'x'. Reorder the terms: 0.75 + -20 + 2x + x2 = 20 + -20 Combine like terms: 0.75 + -20 = -19.25 -19.25 + 2x + x2 = 20 + -20 Combine like terms: 20 + -20 = 0 -19.25 + 2x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '19.25' to each side of the equation. -19.25 + 2x + 19.25 + x2 = 0 + 19.25 Reorder the terms: -19.25 + 19.25 + 2x + x2 = 0 + 19.25 Combine like terms: -19.25 + 19.25 = 0.00 0.00 + 2x + x2 = 0 + 19.25 2x + x2 = 0 + 19.25 Combine like terms: 0 + 19.25 = 19.25 2x + x2 = 19.25 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 = 19.25 + 1 Reorder the terms: 1 + 2x + x2 = 19.25 + 1 Combine like terms: 19.25 + 1 = 20.25 1 + 2x + x2 = 20.25 Factor a perfect square on the left side: (x + 1)(x + 1) = 20.25 Calculate the square root of the right side: 4.5 Break this problem into two subproblems by setting (x + 1) equal to 4.5 and -4.5.Subproblem 1
x + 1 = 4.5 Simplifying x + 1 = 4.5 Reorder the terms: 1 + x = 4.5 Solving 1 + x = 4.5 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 = 4.5 + -1 Combine like terms: 1 + -1 = 0 0 + x = 4.5 + -1 x = 4.5 + -1 Combine like terms: 4.5 + -1 = 3.5 x = 3.5 Simplifying x = 3.5Subproblem 2
x + 1 = -4.5 Simplifying x + 1 = -4.5 Reorder the terms: 1 + x = -4.5 Solving 1 + x = -4.5 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 = -4.5 + -1 Combine like terms: 1 + -1 = 0 0 + x = -4.5 + -1 x = -4.5 + -1 Combine like terms: -4.5 + -1 = -5.5 x = -5.5 Simplifying x = -5.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.5, -5.5}
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