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Simplifying 0.5x + 1[(5x + -4)(3x + 2)] = 0 Reorder the terms: 0.5x + 1[(-4 + 5x)(3x + 2)] = 0 Reorder the terms: 0.5x + 1[(-4 + 5x)(2 + 3x)] = 0 Multiply (-4 + 5x) * (2 + 3x) 0.5x + 1[(-4(2 + 3x) + 5x * (2 + 3x))] = 0 0.5x + 1[((2 * -4 + 3x * -4) + 5x * (2 + 3x))] = 0 0.5x + 1[((-8 + -12x) + 5x * (2 + 3x))] = 0 0.5x + 1[(-8 + -12x + (2 * 5x + 3x * 5x))] = 0 0.5x + 1[(-8 + -12x + (10x + 15x2))] = 0 Combine like terms: -12x + 10x = -2x 0.5x + 1[(-8 + -2x + 15x2)] = 0 0.5x + [-8 * 1 + -2x * 1 + 15x2 * 1] = 0 0.5x + [-8 + -2x + 15x2] = 0 Reorder the terms: -8 + 0.5x + -2x + 15x2 = 0 Combine like terms: 0.5x + -2x = -1.5x -8 + -1.5x + 15x2 = 0 Solving -8 + -1.5x + 15x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.5333333333 + -0.1x + x2 = 0 Move the constant term to the right: Add '0.5333333333' to each side of the equation. -0.5333333333 + -0.1x + 0.5333333333 + x2 = 0 + 0.5333333333 Reorder the terms: -0.5333333333 + 0.5333333333 + -0.1x + x2 = 0 + 0.5333333333 Combine like terms: -0.5333333333 + 0.5333333333 = 0.0000000000 0.0000000000 + -0.1x + x2 = 0 + 0.5333333333 -0.1x + x2 = 0 + 0.5333333333 Combine like terms: 0 + 0.5333333333 = 0.5333333333 -0.1x + x2 = 0.5333333333 The x term is -0.1x. Take half its coefficient (-0.05). Square it (0.0025) and add it to both sides. Add '0.0025' to each side of the equation. -0.1x + 0.0025 + x2 = 0.5333333333 + 0.0025 Reorder the terms: 0.0025 + -0.1x + x2 = 0.5333333333 + 0.0025 Combine like terms: 0.5333333333 + 0.0025 = 0.5358333333 0.0025 + -0.1x + x2 = 0.5358333333 Factor a perfect square on the left side: (x + -0.05)(x + -0.05) = 0.5358333333 Calculate the square root of the right side: 0.732006375 Break this problem into two subproblems by setting (x + -0.05) equal to 0.732006375 and -0.732006375.Subproblem 1
x + -0.05 = 0.732006375 Simplifying x + -0.05 = 0.732006375 Reorder the terms: -0.05 + x = 0.732006375 Solving -0.05 + x = 0.732006375 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + x = 0.732006375 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + x = 0.732006375 + 0.05 x = 0.732006375 + 0.05 Combine like terms: 0.732006375 + 0.05 = 0.782006375 x = 0.782006375 Simplifying x = 0.782006375Subproblem 2
x + -0.05 = -0.732006375 Simplifying x + -0.05 = -0.732006375 Reorder the terms: -0.05 + x = -0.732006375 Solving -0.05 + x = -0.732006375 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + x = -0.732006375 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + x = -0.732006375 + 0.05 x = -0.732006375 + 0.05 Combine like terms: -0.732006375 + 0.05 = -0.682006375 x = -0.682006375 Simplifying x = -0.682006375Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.782006375, -0.682006375}
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