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Simplifying 7k2 + -6k + -3 = 3 Reorder the terms: -3 + -6k + 7k2 = 3 Solving -3 + -6k + 7k2 = 3 Solving for variable 'k'. Reorder the terms: -3 + -3 + -6k + 7k2 = 3 + -3 Combine like terms: -3 + -3 = -6 -6 + -6k + 7k2 = 3 + -3 Combine like terms: 3 + -3 = 0 -6 + -6k + 7k2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.8571428571 + -0.8571428571k + k2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + -0.8571428571k + 0.8571428571 + k2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + -0.8571428571k + k2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + -0.8571428571k + k2 = 0 + 0.8571428571 -0.8571428571k + k2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 -0.8571428571k + k2 = 0.8571428571 The k term is -0.8571428571k. 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. -0.8571428571k + 0.1836734694 + k2 = 0.8571428571 + 0.1836734694 Reorder the terms: 0.1836734694 + -0.8571428571k + k2 = 0.8571428571 + 0.1836734694 Combine like terms: 0.8571428571 + 0.1836734694 = 1.0408163265 0.1836734694 + -0.8571428571k + k2 = 1.0408163265 Factor a perfect square on the left side: (k + -0.4285714286)(k + -0.4285714286) = 1.0408163265 Calculate the square root of the right side: 1.020204061 Break this problem into two subproblems by setting (k + -0.4285714286) equal to 1.020204061 and -1.020204061.Subproblem 1
k + -0.4285714286 = 1.020204061 Simplifying k + -0.4285714286 = 1.020204061 Reorder the terms: -0.4285714286 + k = 1.020204061 Solving -0.4285714286 + k = 1.020204061 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + k = 1.020204061 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + k = 1.020204061 + 0.4285714286 k = 1.020204061 + 0.4285714286 Combine like terms: 1.020204061 + 0.4285714286 = 1.4487754896 k = 1.4487754896 Simplifying k = 1.4487754896Subproblem 2
k + -0.4285714286 = -1.020204061 Simplifying k + -0.4285714286 = -1.020204061 Reorder the terms: -0.4285714286 + k = -1.020204061 Solving -0.4285714286 + k = -1.020204061 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + k = -1.020204061 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + k = -1.020204061 + 0.4285714286 k = -1.020204061 + 0.4285714286 Combine like terms: -1.020204061 + 0.4285714286 = -0.5916326324 k = -0.5916326324 Simplifying k = -0.5916326324Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.4487754896, -0.5916326324}
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