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Simplifying (5x + 3)(3x + 7) = 180 Reorder the terms: (3 + 5x)(3x + 7) = 180 Reorder the terms: (3 + 5x)(7 + 3x) = 180 Multiply (3 + 5x) * (7 + 3x) (3(7 + 3x) + 5x * (7 + 3x)) = 180 ((7 * 3 + 3x * 3) + 5x * (7 + 3x)) = 180 ((21 + 9x) + 5x * (7 + 3x)) = 180 (21 + 9x + (7 * 5x + 3x * 5x)) = 180 (21 + 9x + (35x + 15x2)) = 180 Combine like terms: 9x + 35x = 44x (21 + 44x + 15x2) = 180 Solving 21 + 44x + 15x2 = 180 Solving for variable 'x'. Reorder the terms: 21 + -180 + 44x + 15x2 = 180 + -180 Combine like terms: 21 + -180 = -159 -159 + 44x + 15x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -159 + 44x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -10.6 + 2.933333333x + x2 = 0 Move the constant term to the right: Add '10.6' to each side of the equation. -10.6 + 2.933333333x + 10.6 + x2 = 0 + 10.6 Reorder the terms: -10.6 + 10.6 + 2.933333333x + x2 = 0 + 10.6 Combine like terms: -10.6 + 10.6 = 0.0 0.0 + 2.933333333x + x2 = 0 + 10.6 2.933333333x + x2 = 0 + 10.6 Combine like terms: 0 + 10.6 = 10.6 2.933333333x + x2 = 10.6 The x term is 2.933333333x. Take half its coefficient (1.466666667). Square it (2.151111112) and add it to both sides. Add '2.151111112' to each side of the equation. 2.933333333x + 2.151111112 + x2 = 10.6 + 2.151111112 Reorder the terms: 2.151111112 + 2.933333333x + x2 = 10.6 + 2.151111112 Combine like terms: 10.6 + 2.151111112 = 12.751111112 2.151111112 + 2.933333333x + x2 = 12.751111112 Factor a perfect square on the left side: (x + 1.466666667)(x + 1.466666667) = 12.751111112 Calculate the square root of the right side: 3.570869798 Break this problem into two subproblems by setting (x + 1.466666667) equal to 3.570869798 and -3.570869798.Subproblem 1
x + 1.466666667 = 3.570869798 Simplifying x + 1.466666667 = 3.570869798 Reorder the terms: 1.466666667 + x = 3.570869798 Solving 1.466666667 + x = 3.570869798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.466666667' to each side of the equation. 1.466666667 + -1.466666667 + x = 3.570869798 + -1.466666667 Combine like terms: 1.466666667 + -1.466666667 = 0.000000000 0.000000000 + x = 3.570869798 + -1.466666667 x = 3.570869798 + -1.466666667 Combine like terms: 3.570869798 + -1.466666667 = 2.104203131 x = 2.104203131 Simplifying x = 2.104203131Subproblem 2
x + 1.466666667 = -3.570869798 Simplifying x + 1.466666667 = -3.570869798 Reorder the terms: 1.466666667 + x = -3.570869798 Solving 1.466666667 + x = -3.570869798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.466666667' to each side of the equation. 1.466666667 + -1.466666667 + x = -3.570869798 + -1.466666667 Combine like terms: 1.466666667 + -1.466666667 = 0.000000000 0.000000000 + x = -3.570869798 + -1.466666667 x = -3.570869798 + -1.466666667 Combine like terms: -3.570869798 + -1.466666667 = -5.037536465 x = -5.037536465 Simplifying x = -5.037536465Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.104203131, -5.037536465}
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