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Simplifying -4(1 + 8k) + 7k(7k + -3) = 0 (1 * -4 + 8k * -4) + 7k(7k + -3) = 0 (-4 + -32k) + 7k(7k + -3) = 0 Reorder the terms: -4 + -32k + 7k(-3 + 7k) = 0 -4 + -32k + (-3 * 7k + 7k * 7k) = 0 -4 + -32k + (-21k + 49k2) = 0 Combine like terms: -32k + -21k = -53k -4 + -53k + 49k2 = 0 Solving -4 + -53k + 49k2 = 0 Solving for variable 'k'. Begin completing the square. Divide all terms by 49 the coefficient of the squared term: Divide each side by '49'. -0.08163265306 + -1.081632653k + k2 = 0 Move the constant term to the right: Add '0.08163265306' to each side of the equation. -0.08163265306 + -1.081632653k + 0.08163265306 + k2 = 0 + 0.08163265306 Reorder the terms: -0.08163265306 + 0.08163265306 + -1.081632653k + k2 = 0 + 0.08163265306 Combine like terms: -0.08163265306 + 0.08163265306 = 0.00000000000 0.00000000000 + -1.081632653k + k2 = 0 + 0.08163265306 -1.081632653k + k2 = 0 + 0.08163265306 Combine like terms: 0 + 0.08163265306 = 0.08163265306 -1.081632653k + k2 = 0.08163265306 The k term is -1.081632653k. Take half its coefficient (-0.5408163265). Square it (0.2924822990) and add it to both sides. Add '0.2924822990' to each side of the equation. -1.081632653k + 0.2924822990 + k2 = 0.08163265306 + 0.2924822990 Reorder the terms: 0.2924822990 + -1.081632653k + k2 = 0.08163265306 + 0.2924822990 Combine like terms: 0.08163265306 + 0.2924822990 = 0.37411495206 0.2924822990 + -1.081632653k + k2 = 0.37411495206 Factor a perfect square on the left side: (k + -0.5408163265)(k + -0.5408163265) = 0.37411495206 Calculate the square root of the right side: 0.61164937 Break this problem into two subproblems by setting (k + -0.5408163265) equal to 0.61164937 and -0.61164937.Subproblem 1
k + -0.5408163265 = 0.61164937 Simplifying k + -0.5408163265 = 0.61164937 Reorder the terms: -0.5408163265 + k = 0.61164937 Solving -0.5408163265 + k = 0.61164937 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5408163265' to each side of the equation. -0.5408163265 + 0.5408163265 + k = 0.61164937 + 0.5408163265 Combine like terms: -0.5408163265 + 0.5408163265 = 0.0000000000 0.0000000000 + k = 0.61164937 + 0.5408163265 k = 0.61164937 + 0.5408163265 Combine like terms: 0.61164937 + 0.5408163265 = 1.1524656965 k = 1.1524656965 Simplifying k = 1.1524656965Subproblem 2
k + -0.5408163265 = -0.61164937 Simplifying k + -0.5408163265 = -0.61164937 Reorder the terms: -0.5408163265 + k = -0.61164937 Solving -0.5408163265 + k = -0.61164937 Solving for variable 'k'. Move all terms containing k to the left, all other terms to the right. Add '0.5408163265' to each side of the equation. -0.5408163265 + 0.5408163265 + k = -0.61164937 + 0.5408163265 Combine like terms: -0.5408163265 + 0.5408163265 = 0.0000000000 0.0000000000 + k = -0.61164937 + 0.5408163265 k = -0.61164937 + 0.5408163265 Combine like terms: -0.61164937 + 0.5408163265 = -0.0708330435 k = -0.0708330435 Simplifying k = -0.0708330435Solution
The solution to the problem is based on the solutions from the subproblems. k = {1.1524656965, -0.0708330435}
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