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Simplifying (x + -5)(1 + -3x) = 7 Reorder the terms: (-5 + x)(1 + -3x) = 7 Multiply (-5 + x) * (1 + -3x) (-5(1 + -3x) + x(1 + -3x)) = 7 ((1 * -5 + -3x * -5) + x(1 + -3x)) = 7 ((-5 + 15x) + x(1 + -3x)) = 7 (-5 + 15x + (1 * x + -3x * x)) = 7 (-5 + 15x + (1x + -3x2)) = 7 Combine like terms: 15x + 1x = 16x (-5 + 16x + -3x2) = 7 Solving -5 + 16x + -3x2 = 7 Solving for variable 'x'. Reorder the terms: -5 + -7 + 16x + -3x2 = 7 + -7 Combine like terms: -5 + -7 = -12 -12 + 16x + -3x2 = 7 + -7 Combine like terms: 7 + -7 = 0 -12 + 16x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 4 + -5.333333333x + x2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -5.333333333x + -4 + x2 = 0 + -4 Reorder the terms: 4 + -4 + -5.333333333x + x2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -5.333333333x + x2 = 0 + -4 -5.333333333x + x2 = 0 + -4 Combine like terms: 0 + -4 = -4 -5.333333333x + x2 = -4 The x term is -5.333333333x. Take half its coefficient (-2.666666667). Square it (7.111111113) and add it to both sides. Add '7.111111113' to each side of the equation. -5.333333333x + 7.111111113 + x2 = -4 + 7.111111113 Reorder the terms: 7.111111113 + -5.333333333x + x2 = -4 + 7.111111113 Combine like terms: -4 + 7.111111113 = 3.111111113 7.111111113 + -5.333333333x + x2 = 3.111111113 Factor a perfect square on the left side: (x + -2.666666667)(x + -2.666666667) = 3.111111113 Calculate the square root of the right side: 1.763834208 Break this problem into two subproblems by setting (x + -2.666666667) equal to 1.763834208 and -1.763834208.Subproblem 1
x + -2.666666667 = 1.763834208 Simplifying x + -2.666666667 = 1.763834208 Reorder the terms: -2.666666667 + x = 1.763834208 Solving -2.666666667 + x = 1.763834208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x = 1.763834208 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x = 1.763834208 + 2.666666667 x = 1.763834208 + 2.666666667 Combine like terms: 1.763834208 + 2.666666667 = 4.430500875 x = 4.430500875 Simplifying x = 4.430500875Subproblem 2
x + -2.666666667 = -1.763834208 Simplifying x + -2.666666667 = -1.763834208 Reorder the terms: -2.666666667 + x = -1.763834208 Solving -2.666666667 + x = -1.763834208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.666666667' to each side of the equation. -2.666666667 + 2.666666667 + x = -1.763834208 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + x = -1.763834208 + 2.666666667 x = -1.763834208 + 2.666666667 Combine like terms: -1.763834208 + 2.666666667 = 0.902832459 x = 0.902832459 Simplifying x = 0.902832459Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.430500875, 0.902832459}
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