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Simplifying 2x(6 + 7x) = 180 (6 * 2x + 7x * 2x) = 180 (12x + 14x2) = 180 Solving 12x + 14x2 = 180 Solving for variable 'x'. Reorder the terms: -180 + 12x + 14x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 12x + 14x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-90 + 6x + 7x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-90 + 6x + 7x2)' equal to zero and attempt to solve: Simplifying -90 + 6x + 7x2 = 0 Solving -90 + 6x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -12.85714286 + 0.8571428571x + x2 = 0 Move the constant term to the right: Add '12.85714286' to each side of the equation. -12.85714286 + 0.8571428571x + 12.85714286 + x2 = 0 + 12.85714286 Reorder the terms: -12.85714286 + 12.85714286 + 0.8571428571x + x2 = 0 + 12.85714286 Combine like terms: -12.85714286 + 12.85714286 = 0.00000000 0.00000000 + 0.8571428571x + x2 = 0 + 12.85714286 0.8571428571x + x2 = 0 + 12.85714286 Combine like terms: 0 + 12.85714286 = 12.85714286 0.8571428571x + x2 = 12.85714286 The x term is 0.8571428571x. Take half its coefficient (0.4285714286). Square it (0.1836734694) and add it to both sides. Add '0.1836734694' to each side of the equation. 0.8571428571x + 0.1836734694 + x2 = 12.85714286 + 0.1836734694 Reorder the terms: 0.1836734694 + 0.8571428571x + x2 = 12.85714286 + 0.1836734694 Combine like terms: 12.85714286 + 0.1836734694 = 13.0408163294 0.1836734694 + 0.8571428571x + x2 = 13.0408163294 Factor a perfect square on the left side: (x + 0.4285714286)(x + 0.4285714286) = 13.0408163294 Calculate the square root of the right side: 3.611207046 Break this problem into two subproblems by setting (x + 0.4285714286) equal to 3.611207046 and -3.611207046.Subproblem 1
x + 0.4285714286 = 3.611207046 Simplifying x + 0.4285714286 = 3.611207046 Reorder the terms: 0.4285714286 + x = 3.611207046 Solving 0.4285714286 + x = 3.611207046 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + x = 3.611207046 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + x = 3.611207046 + -0.4285714286 x = 3.611207046 + -0.4285714286 Combine like terms: 3.611207046 + -0.4285714286 = 3.1826356174 x = 3.1826356174 Simplifying x = 3.1826356174Subproblem 2
x + 0.4285714286 = -3.611207046 Simplifying x + 0.4285714286 = -3.611207046 Reorder the terms: 0.4285714286 + x = -3.611207046 Solving 0.4285714286 + x = -3.611207046 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + x = -3.611207046 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + x = -3.611207046 + -0.4285714286 x = -3.611207046 + -0.4285714286 Combine like terms: -3.611207046 + -0.4285714286 = -4.0397784746 x = -4.0397784746 Simplifying x = -4.0397784746Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.1826356174, -4.0397784746}Solution
x = {3.1826356174, -4.0397784746}
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