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Simplifying 0 = 21(0.25 + -1x)(0.55 + -1x) Multiply (0.25 + -1x) * (0.55 + -1x) 0 = 21(0.25(0.55 + -1x) + -1x * (0.55 + -1x)) 0 = 21((0.55 * 0.25 + -1x * 0.25) + -1x * (0.55 + -1x)) 0 = 21((0.1375 + -0.25x) + -1x * (0.55 + -1x)) 0 = 21(0.1375 + -0.25x + (0.55 * -1x + -1x * -1x)) 0 = 21(0.1375 + -0.25x + (-0.55x + 1x2)) Combine like terms: -0.25x + -0.55x = -0.8x 0 = 21(0.1375 + -0.8x + 1x2) 0 = (0.1375 * 21 + -0.8x * 21 + 1x2 * 21) 0 = (2.8875 + -16.8x + 21x2) Solving 0 = 2.8875 + -16.8x + 21x2 Solving for variable 'x'. Combine like terms: 0 + -2.8875 = -2.8875 -2.8875 + 16.8x + -21x2 = 2.8875 + -16.8x + 21x2 + -2.8875 + 16.8x + -21x2 Reorder the terms: -2.8875 + 16.8x + -21x2 = 2.8875 + -2.8875 + -16.8x + 16.8x + 21x2 + -21x2 Combine like terms: 2.8875 + -2.8875 = 0.0000 -2.8875 + 16.8x + -21x2 = 0.0000 + -16.8x + 16.8x + 21x2 + -21x2 -2.8875 + 16.8x + -21x2 = -16.8x + 16.8x + 21x2 + -21x2 Combine like terms: -16.8x + 16.8x = 0.0 -2.8875 + 16.8x + -21x2 = 0.0 + 21x2 + -21x2 -2.8875 + 16.8x + -21x2 = 21x2 + -21x2 Combine like terms: 21x2 + -21x2 = 0 -2.8875 + 16.8x + -21x2 = 0 Begin completing the square. Divide all terms by -21 the coefficient of the squared term: Divide each side by '-21'. 0.1375 + -0.8x + x2 = 0 Move the constant term to the right: Add '-0.1375' to each side of the equation. 0.1375 + -0.8x + -0.1375 + x2 = 0 + -0.1375 Reorder the terms: 0.1375 + -0.1375 + -0.8x + x2 = 0 + -0.1375 Combine like terms: 0.1375 + -0.1375 = 0.0000 0.0000 + -0.8x + x2 = 0 + -0.1375 -0.8x + x2 = 0 + -0.1375 Combine like terms: 0 + -0.1375 = -0.1375 -0.8x + x2 = -0.1375 The x term is -0.8x. Take half its coefficient (-0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. -0.8x + 0.16 + x2 = -0.1375 + 0.16 Reorder the terms: 0.16 + -0.8x + x2 = -0.1375 + 0.16 Combine like terms: -0.1375 + 0.16 = 0.0225 0.16 + -0.8x + x2 = 0.0225 Factor a perfect square on the left side: (x + -0.4)(x + -0.4) = 0.0225 Calculate the square root of the right side: 0.15 Break this problem into two subproblems by setting (x + -0.4) equal to 0.15 and -0.15.Subproblem 1
x + -0.4 = 0.15 Simplifying x + -0.4 = 0.15 Reorder the terms: -0.4 + x = 0.15 Solving -0.4 + x = 0.15 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + x = 0.15 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + x = 0.15 + 0.4 x = 0.15 + 0.4 Combine like terms: 0.15 + 0.4 = 0.55 x = 0.55 Simplifying x = 0.55Subproblem 2
x + -0.4 = -0.15 Simplifying x + -0.4 = -0.15 Reorder the terms: -0.4 + x = -0.15 Solving -0.4 + x = -0.15 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + x = -0.15 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + x = -0.15 + 0.4 x = -0.15 + 0.4 Combine like terms: -0.15 + 0.4 = 0.25 x = 0.25 Simplifying x = 0.25Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.55, 0.25}
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