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Simplifying 9 + -1(6x + 1) * x = 7.5 Reorder the terms: 9 + -1(1 + 6x) * x = 7.5 Reorder the terms for easier multiplication: 9 + -1x(1 + 6x) = 7.5 9 + (1 * -1x + 6x * -1x) = 7.5 9 + (-1x + -6x2) = 7.5 Solving 9 + -1x + -6x2 = 7.5 Solving for variable 'x'. Reorder the terms: 9 + -7.5 + -1x + -6x2 = 7.5 + -7.5 Combine like terms: 9 + -7.5 = 1.5 1.5 + -1x + -6x2 = 7.5 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 1.5 + -1x + -6x2 = 0.0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -0.25 + 0.1666666667x + x2 = 0.0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 0.1666666667x + 0.25 + x2 = 0.0 + 0.25 Reorder the terms: -0.25 + 0.25 + 0.1666666667x + x2 = 0.0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 0.1666666667x + x2 = 0.0 + 0.25 0.1666666667x + x2 = 0.0 + 0.25 Combine like terms: 0.0 + 0.25 = 0.25 0.1666666667x + x2 = 0.25 The x term is 0.1666666667x. Take half its coefficient (0.08333333335). Square it (0.006944444447) and add it to both sides. Add '0.006944444447' to each side of the equation. 0.1666666667x + 0.006944444447 + x2 = 0.25 + 0.006944444447 Reorder the terms: 0.006944444447 + 0.1666666667x + x2 = 0.25 + 0.006944444447 Combine like terms: 0.25 + 0.006944444447 = 0.256944444447 0.006944444447 + 0.1666666667x + x2 = 0.256944444447 Factor a perfect square on the left side: (x + 0.08333333335)(x + 0.08333333335) = 0.256944444447 Calculate the square root of the right side: 0.506896878 Break this problem into two subproblems by setting (x + 0.08333333335) equal to 0.506896878 and -0.506896878.Subproblem 1
x + 0.08333333335 = 0.506896878 Simplifying x + 0.08333333335 = 0.506896878 Reorder the terms: 0.08333333335 + x = 0.506896878 Solving 0.08333333335 + x = 0.506896878 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = 0.506896878 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = 0.506896878 + -0.08333333335 x = 0.506896878 + -0.08333333335 Combine like terms: 0.506896878 + -0.08333333335 = 0.42356354465 x = 0.42356354465 Simplifying x = 0.42356354465Subproblem 2
x + 0.08333333335 = -0.506896878 Simplifying x + 0.08333333335 = -0.506896878 Reorder the terms: 0.08333333335 + x = -0.506896878 Solving 0.08333333335 + x = -0.506896878 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = -0.506896878 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = -0.506896878 + -0.08333333335 x = -0.506896878 + -0.08333333335 Combine like terms: -0.506896878 + -0.08333333335 = -0.59023021135 x = -0.59023021135 Simplifying x = -0.59023021135Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.42356354465, -0.59023021135}
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