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Simplifying 6x(6 + 7x) = 45 (6 * 6x + 7x * 6x) = 45 (36x + 42x2) = 45 Solving 36x + 42x2 = 45 Solving for variable 'x'. Reorder the terms: -45 + 36x + 42x2 = 45 + -45 Combine like terms: 45 + -45 = 0 -45 + 36x + 42x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-15 + 12x + 14x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-15 + 12x + 14x2)' equal to zero and attempt to solve: Simplifying -15 + 12x + 14x2 = 0 Solving -15 + 12x + 14x2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -1.071428571 + 0.8571428571x + x2 = 0 Move the constant term to the right: Add '1.071428571' to each side of the equation. -1.071428571 + 0.8571428571x + 1.071428571 + x2 = 0 + 1.071428571 Reorder the terms: -1.071428571 + 1.071428571 + 0.8571428571x + x2 = 0 + 1.071428571 Combine like terms: -1.071428571 + 1.071428571 = 0.000000000 0.000000000 + 0.8571428571x + x2 = 0 + 1.071428571 0.8571428571x + x2 = 0 + 1.071428571 Combine like terms: 0 + 1.071428571 = 1.071428571 0.8571428571x + x2 = 1.071428571 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 = 1.071428571 + 0.1836734694 Reorder the terms: 0.1836734694 + 0.8571428571x + x2 = 1.071428571 + 0.1836734694 Combine like terms: 1.071428571 + 0.1836734694 = 1.2551020404 0.1836734694 + 0.8571428571x + x2 = 1.2551020404 Factor a perfect square on the left side: (x + 0.4285714286)(x + 0.4285714286) = 1.2551020404 Calculate the square root of the right side: 1.120313367 Break this problem into two subproblems by setting (x + 0.4285714286) equal to 1.120313367 and -1.120313367.Subproblem 1
x + 0.4285714286 = 1.120313367 Simplifying x + 0.4285714286 = 1.120313367 Reorder the terms: 0.4285714286 + x = 1.120313367 Solving 0.4285714286 + x = 1.120313367 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 = 1.120313367 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + x = 1.120313367 + -0.4285714286 x = 1.120313367 + -0.4285714286 Combine like terms: 1.120313367 + -0.4285714286 = 0.6917419384 x = 0.6917419384 Simplifying x = 0.6917419384Subproblem 2
x + 0.4285714286 = -1.120313367 Simplifying x + 0.4285714286 = -1.120313367 Reorder the terms: 0.4285714286 + x = -1.120313367 Solving 0.4285714286 + x = -1.120313367 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 = -1.120313367 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + x = -1.120313367 + -0.4285714286 x = -1.120313367 + -0.4285714286 Combine like terms: -1.120313367 + -0.4285714286 = -1.5488847956 x = -1.5488847956 Simplifying x = -1.5488847956Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.6917419384, -1.5488847956}Solution
x = {0.6917419384, -1.5488847956}
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