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Simplifying 4x2 = (0.25 + -1x)(0.25 + -1x)(7.0) Reorder the terms for easier multiplication: 4x2 = 7.0(0.25 + -1x)(0.25 + -1x) Multiply (0.25 + -1x) * (0.25 + -1x) 4x2 = 7.0(0.25(0.25 + -1x) + -1x * (0.25 + -1x)) 4x2 = 7.0((0.25 * 0.25 + -1x * 0.25) + -1x * (0.25 + -1x)) 4x2 = 7.0((0.0625 + -0.25x) + -1x * (0.25 + -1x)) 4x2 = 7.0(0.0625 + -0.25x + (0.25 * -1x + -1x * -1x)) 4x2 = 7.0(0.0625 + -0.25x + (-0.25x + 1x2)) Combine like terms: -0.25x + -0.25x = -0.5x 4x2 = 7.0(0.0625 + -0.5x + 1x2) 4x2 = (0.0625 * 7.0 + -0.5x * 7.0 + 1x2 * 7.0) 4x2 = (0.4375 + -3.5x + 7x2) Solving 4x2 = 0.4375 + -3.5x + 7x2 Solving for variable 'x'. Reorder the terms: -0.4375 + 3.5x + 4x2 + -7x2 = 0.4375 + -3.5x + 7x2 + -0.4375 + 3.5x + -7x2 Combine like terms: 4x2 + -7x2 = -3x2 -0.4375 + 3.5x + -3x2 = 0.4375 + -3.5x + 7x2 + -0.4375 + 3.5x + -7x2 Reorder the terms: -0.4375 + 3.5x + -3x2 = 0.4375 + -0.4375 + -3.5x + 3.5x + 7x2 + -7x2 Combine like terms: 0.4375 + -0.4375 = 0.0000 -0.4375 + 3.5x + -3x2 = 0.0000 + -3.5x + 3.5x + 7x2 + -7x2 -0.4375 + 3.5x + -3x2 = -3.5x + 3.5x + 7x2 + -7x2 Combine like terms: -3.5x + 3.5x = 0.0 -0.4375 + 3.5x + -3x2 = 0.0 + 7x2 + -7x2 -0.4375 + 3.5x + -3x2 = 7x2 + -7x2 Combine like terms: 7x2 + -7x2 = 0 -0.4375 + 3.5x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 0.1458333333 + -1.166666667x + x2 = 0 Move the constant term to the right: Add '-0.1458333333' to each side of the equation. 0.1458333333 + -1.166666667x + -0.1458333333 + x2 = 0 + -0.1458333333 Reorder the terms: 0.1458333333 + -0.1458333333 + -1.166666667x + x2 = 0 + -0.1458333333 Combine like terms: 0.1458333333 + -0.1458333333 = 0.0000000000 0.0000000000 + -1.166666667x + x2 = 0 + -0.1458333333 -1.166666667x + x2 = 0 + -0.1458333333 Combine like terms: 0 + -0.1458333333 = -0.1458333333 -1.166666667x + x2 = -0.1458333333 The x term is -1.166666667x. Take half its coefficient (-0.5833333335). Square it (0.3402777780) and add it to both sides. Add '0.3402777780' to each side of the equation. -1.166666667x + 0.3402777780 + x2 = -0.1458333333 + 0.3402777780 Reorder the terms: 0.3402777780 + -1.166666667x + x2 = -0.1458333333 + 0.3402777780 Combine like terms: -0.1458333333 + 0.3402777780 = 0.1944444447 0.3402777780 + -1.166666667x + x2 = 0.1944444447 Factor a perfect square on the left side: (x + -0.5833333335)(x + -0.5833333335) = 0.1944444447 Calculate the square root of the right side: 0.440958552 Break this problem into two subproblems by setting (x + -0.5833333335) equal to 0.440958552 and -0.440958552.Subproblem 1
x + -0.5833333335 = 0.440958552 Simplifying x + -0.5833333335 = 0.440958552 Reorder the terms: -0.5833333335 + x = 0.440958552 Solving -0.5833333335 + x = 0.440958552 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + x = 0.440958552 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = 0.440958552 + 0.5833333335 x = 0.440958552 + 0.5833333335 Combine like terms: 0.440958552 + 0.5833333335 = 1.0242918855 x = 1.0242918855 Simplifying x = 1.0242918855Subproblem 2
x + -0.5833333335 = -0.440958552 Simplifying x + -0.5833333335 = -0.440958552 Reorder the terms: -0.5833333335 + x = -0.440958552 Solving -0.5833333335 + x = -0.440958552 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + x = -0.440958552 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = -0.440958552 + 0.5833333335 x = -0.440958552 + 0.5833333335 Combine like terms: -0.440958552 + 0.5833333335 = 0.1423747815 x = 0.1423747815 Simplifying x = 0.1423747815Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.0242918855, 0.1423747815}
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