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Simplifying 6(x + 7) + 3 = 11x(x + -5) Reorder the terms: 6(7 + x) + 3 = 11x(x + -5) (7 * 6 + x * 6) + 3 = 11x(x + -5) (42 + 6x) + 3 = 11x(x + -5) Reorder the terms: 42 + 3 + 6x = 11x(x + -5) Combine like terms: 42 + 3 = 45 45 + 6x = 11x(x + -5) Reorder the terms: 45 + 6x = 11x(-5 + x) 45 + 6x = (-5 * 11x + x * 11x) 45 + 6x = (-55x + 11x2) Solving 45 + 6x = -55x + 11x2 Solving for variable 'x'. Combine like terms: 6x + 55x = 61x 45 + 61x + -11x2 = -55x + 11x2 + 55x + -11x2 Reorder the terms: 45 + 61x + -11x2 = -55x + 55x + 11x2 + -11x2 Combine like terms: -55x + 55x = 0 45 + 61x + -11x2 = 0 + 11x2 + -11x2 45 + 61x + -11x2 = 11x2 + -11x2 Combine like terms: 11x2 + -11x2 = 0 45 + 61x + -11x2 = 0 Begin completing the square. Divide all terms by -11 the coefficient of the squared term: Divide each side by '-11'. -4.090909091 + -5.545454545x + x2 = 0 Move the constant term to the right: Add '4.090909091' to each side of the equation. -4.090909091 + -5.545454545x + 4.090909091 + x2 = 0 + 4.090909091 Reorder the terms: -4.090909091 + 4.090909091 + -5.545454545x + x2 = 0 + 4.090909091 Combine like terms: -4.090909091 + 4.090909091 = 0.000000000 0.000000000 + -5.545454545x + x2 = 0 + 4.090909091 -5.545454545x + x2 = 0 + 4.090909091 Combine like terms: 0 + 4.090909091 = 4.090909091 -5.545454545x + x2 = 4.090909091 The x term is -5.545454545x. Take half its coefficient (-2.772727273). Square it (7.688016530) and add it to both sides. Add '7.688016530' to each side of the equation. -5.545454545x + 7.688016530 + x2 = 4.090909091 + 7.688016530 Reorder the terms: 7.688016530 + -5.545454545x + x2 = 4.090909091 + 7.688016530 Combine like terms: 4.090909091 + 7.688016530 = 11.778925621 7.688016530 + -5.545454545x + x2 = 11.778925621 Factor a perfect square on the left side: (x + -2.772727273)(x + -2.772727273) = 11.778925621 Calculate the square root of the right side: 3.432043942 Break this problem into two subproblems by setting (x + -2.772727273) equal to 3.432043942 and -3.432043942.Subproblem 1
x + -2.772727273 = 3.432043942 Simplifying x + -2.772727273 = 3.432043942 Reorder the terms: -2.772727273 + x = 3.432043942 Solving -2.772727273 + x = 3.432043942 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.772727273' to each side of the equation. -2.772727273 + 2.772727273 + x = 3.432043942 + 2.772727273 Combine like terms: -2.772727273 + 2.772727273 = 0.000000000 0.000000000 + x = 3.432043942 + 2.772727273 x = 3.432043942 + 2.772727273 Combine like terms: 3.432043942 + 2.772727273 = 6.204771215 x = 6.204771215 Simplifying x = 6.204771215Subproblem 2
x + -2.772727273 = -3.432043942 Simplifying x + -2.772727273 = -3.432043942 Reorder the terms: -2.772727273 + x = -3.432043942 Solving -2.772727273 + x = -3.432043942 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.772727273' to each side of the equation. -2.772727273 + 2.772727273 + x = -3.432043942 + 2.772727273 Combine like terms: -2.772727273 + 2.772727273 = 0.000000000 0.000000000 + x = -3.432043942 + 2.772727273 x = -3.432043942 + 2.772727273 Combine like terms: -3.432043942 + 2.772727273 = -0.659316669 x = -0.659316669 Simplifying x = -0.659316669Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.204771215, -0.659316669}
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