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Simplifying (6x + 1)(5x + -17)(9 + -24) = 180 Reorder the terms: (1 + 6x)(5x + -17)(9 + -24) = 180 Reorder the terms: (1 + 6x)(-17 + 5x)(9 + -24) = 180 Combine like terms: 9 + -24 = -15 (1 + 6x)(-17 + 5x)(-15) = 180 Reorder the terms for easier multiplication: -15(1 + 6x)(-17 + 5x) = 180 Multiply (1 + 6x) * (-17 + 5x) -15(1(-17 + 5x) + 6x * (-17 + 5x)) = 180 -15((-17 * 1 + 5x * 1) + 6x * (-17 + 5x)) = 180 -15((-17 + 5x) + 6x * (-17 + 5x)) = 180 -15(-17 + 5x + (-17 * 6x + 5x * 6x)) = 180 -15(-17 + 5x + (-102x + 30x2)) = 180 Combine like terms: 5x + -102x = -97x -15(-17 + -97x + 30x2) = 180 (-17 * -15 + -97x * -15 + 30x2 * -15) = 180 (255 + 1455x + -450x2) = 180 Solving 255 + 1455x + -450x2 = 180 Solving for variable 'x'. Reorder the terms: 255 + -180 + 1455x + -450x2 = 180 + -180 Combine like terms: 255 + -180 = 75 75 + 1455x + -450x2 = 180 + -180 Combine like terms: 180 + -180 = 0 75 + 1455x + -450x2 = 0 Factor out the Greatest Common Factor (GCF), '15'. 15(5 + 97x + -30x2) = 0 Ignore the factor 15.Subproblem 1
Set the factor '(5 + 97x + -30x2)' equal to zero and attempt to solve: Simplifying 5 + 97x + -30x2 = 0 Solving 5 + 97x + -30x2 = 0 Begin completing the square. Divide all terms by -30 the coefficient of the squared term: Divide each side by '-30'. -0.1666666667 + -3.233333333x + x2 = 0.0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + -3.233333333x + 0.1666666667 + x2 = 0.0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + -3.233333333x + x2 = 0.0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + -3.233333333x + x2 = 0.0 + 0.1666666667 -3.233333333x + x2 = 0.0 + 0.1666666667 Combine like terms: 0.0 + 0.1666666667 = 0.1666666667 -3.233333333x + x2 = 0.1666666667 The x term is -3.233333333x. Take half its coefficient (-1.616666667). Square it (2.613611112) and add it to both sides. Add '2.613611112' to each side of the equation. -3.233333333x + 2.613611112 + x2 = 0.1666666667 + 2.613611112 Reorder the terms: 2.613611112 + -3.233333333x + x2 = 0.1666666667 + 2.613611112 Combine like terms: 0.1666666667 + 2.613611112 = 2.7802777787 2.613611112 + -3.233333333x + x2 = 2.7802777787 Factor a perfect square on the left side: (x + -1.616666667)(x + -1.616666667) = 2.7802777787 Calculate the square root of the right side: 1.667416498 Break this problem into two subproblems by setting (x + -1.616666667) equal to 1.667416498 and -1.667416498.Subproblem 1
x + -1.616666667 = 1.667416498 Simplifying x + -1.616666667 = 1.667416498 Reorder the terms: -1.616666667 + x = 1.667416498 Solving -1.616666667 + x = 1.667416498 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.616666667' to each side of the equation. -1.616666667 + 1.616666667 + x = 1.667416498 + 1.616666667 Combine like terms: -1.616666667 + 1.616666667 = 0.000000000 0.000000000 + x = 1.667416498 + 1.616666667 x = 1.667416498 + 1.616666667 Combine like terms: 1.667416498 + 1.616666667 = 3.284083165 x = 3.284083165 Simplifying x = 3.284083165Subproblem 2
x + -1.616666667 = -1.667416498 Simplifying x + -1.616666667 = -1.667416498 Reorder the terms: -1.616666667 + x = -1.667416498 Solving -1.616666667 + x = -1.667416498 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.616666667' to each side of the equation. -1.616666667 + 1.616666667 + x = -1.667416498 + 1.616666667 Combine like terms: -1.616666667 + 1.616666667 = 0.000000000 0.000000000 + x = -1.667416498 + 1.616666667 x = -1.667416498 + 1.616666667 Combine like terms: -1.667416498 + 1.616666667 = -0.050749831 x = -0.050749831 Simplifying x = -0.050749831Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.284083165, -0.050749831}Solution
x = {3.284083165, -0.050749831}
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