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Simplifying (2 + -5x)(2 + 5x) = 5(1 + 2x) + -1(5x + 2)(-10x) Multiply (2 + -5x) * (2 + 5x) (2(2 + 5x) + -5x * (2 + 5x)) = 5(1 + 2x) + -1(5x + 2)(-10x) ((2 * 2 + 5x * 2) + -5x * (2 + 5x)) = 5(1 + 2x) + -1(5x + 2)(-10x) ((4 + 10x) + -5x * (2 + 5x)) = 5(1 + 2x) + -1(5x + 2)(-10x) (4 + 10x + (2 * -5x + 5x * -5x)) = 5(1 + 2x) + -1(5x + 2)(-10x) (4 + 10x + (-10x + -25x2)) = 5(1 + 2x) + -1(5x + 2)(-10x) Combine like terms: 10x + -10x = 0 (4 + 0 + -25x2) = 5(1 + 2x) + -1(5x + 2)(-10x) (4 + -25x2) = 5(1 + 2x) + -1(5x + 2)(-10x) 4 + -25x2 = (1 * 5 + 2x * 5) + -1(5x + 2)(-10x) 4 + -25x2 = (5 + 10x) + -1(5x + 2)(-10x) Reorder the terms: 4 + -25x2 = 5 + 10x + -1(2 + 5x)(-10x) Remove parenthesis around (-10x) 4 + -25x2 = 5 + 10x + -1(2 + 5x) * -10x Reorder the terms for easier multiplication: 4 + -25x2 = 5 + 10x + -1 * -10x(2 + 5x) Multiply -1 * -10 4 + -25x2 = 5 + 10x + 10x(2 + 5x) 4 + -25x2 = 5 + 10x + (2 * 10x + 5x * 10x) 4 + -25x2 = 5 + 10x + (20x + 50x2) Combine like terms: 10x + 20x = 30x 4 + -25x2 = 5 + 30x + 50x2 Solving 4 + -25x2 = 5 + 30x + 50x2 Solving for variable 'x'. Reorder the terms: 4 + -5 + -30x + -25x2 + -50x2 = 5 + 30x + 50x2 + -5 + -30x + -50x2 Combine like terms: 4 + -5 = -1 -1 + -30x + -25x2 + -50x2 = 5 + 30x + 50x2 + -5 + -30x + -50x2 Combine like terms: -25x2 + -50x2 = -75x2 -1 + -30x + -75x2 = 5 + 30x + 50x2 + -5 + -30x + -50x2 Reorder the terms: -1 + -30x + -75x2 = 5 + -5 + 30x + -30x + 50x2 + -50x2 Combine like terms: 5 + -5 = 0 -1 + -30x + -75x2 = 0 + 30x + -30x + 50x2 + -50x2 -1 + -30x + -75x2 = 30x + -30x + 50x2 + -50x2 Combine like terms: 30x + -30x = 0 -1 + -30x + -75x2 = 0 + 50x2 + -50x2 -1 + -30x + -75x2 = 50x2 + -50x2 Combine like terms: 50x2 + -50x2 = 0 -1 + -30x + -75x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + 30x + 75x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + 30x + 75x2)' equal to zero and attempt to solve: Simplifying 1 + 30x + 75x2 = 0 Solving 1 + 30x + 75x2 = 0 Begin completing the square. Divide all terms by 75 the coefficient of the squared term: Divide each side by '75'. 0.01333333333 + 0.4x + x2 = 0 Move the constant term to the right: Add '-0.01333333333' to each side of the equation. 0.01333333333 + 0.4x + -0.01333333333 + x2 = 0 + -0.01333333333 Reorder the terms: 0.01333333333 + -0.01333333333 + 0.4x + x2 = 0 + -0.01333333333 Combine like terms: 0.01333333333 + -0.01333333333 = 0.00000000000 0.00000000000 + 0.4x + x2 = 0 + -0.01333333333 0.4x + x2 = 0 + -0.01333333333 Combine like terms: 0 + -0.01333333333 = -0.01333333333 0.4x + x2 = -0.01333333333 The x term is 0.4x. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4x + 0.04 + x2 = -0.01333333333 + 0.04 Reorder the terms: 0.04 + 0.4x + x2 = -0.01333333333 + 0.04 Combine like terms: -0.01333333333 + 0.04 = 0.02666666667 0.04 + 0.4x + x2 = 0.02666666667 Factor a perfect square on the left side: (x + 0.2)(x + 0.2) = 0.02666666667 Calculate the square root of the right side: 0.163299316 Break this problem into two subproblems by setting (x + 0.2) equal to 0.163299316 and -0.163299316.Subproblem 1
x + 0.2 = 0.163299316 Simplifying x + 0.2 = 0.163299316 Reorder the terms: 0.2 + x = 0.163299316 Solving 0.2 + x = 0.163299316 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + x = 0.163299316 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + x = 0.163299316 + -0.2 x = 0.163299316 + -0.2 Combine like terms: 0.163299316 + -0.2 = -0.036700684 x = -0.036700684 Simplifying x = -0.036700684Subproblem 2
x + 0.2 = -0.163299316 Simplifying x + 0.2 = -0.163299316 Reorder the terms: 0.2 + x = -0.163299316 Solving 0.2 + x = -0.163299316 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + x = -0.163299316 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + x = -0.163299316 + -0.2 x = -0.163299316 + -0.2 Combine like terms: -0.163299316 + -0.2 = -0.363299316 x = -0.363299316 Simplifying x = -0.363299316Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.036700684, -0.363299316}Solution
x = {-0.036700684, -0.363299316}
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