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Simplifying 6x(3x + 7) = 29 Reorder the terms: 6x(7 + 3x) = 29 (7 * 6x + 3x * 6x) = 29 (42x + 18x2) = 29 Solving 42x + 18x2 = 29 Solving for variable 'x'. Reorder the terms: -29 + 42x + 18x2 = 29 + -29 Combine like terms: 29 + -29 = 0 -29 + 42x + 18x2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -1.611111111 + 2.333333333x + x2 = 0 Move the constant term to the right: Add '1.611111111' to each side of the equation. -1.611111111 + 2.333333333x + 1.611111111 + x2 = 0 + 1.611111111 Reorder the terms: -1.611111111 + 1.611111111 + 2.333333333x + x2 = 0 + 1.611111111 Combine like terms: -1.611111111 + 1.611111111 = 0.000000000 0.000000000 + 2.333333333x + x2 = 0 + 1.611111111 2.333333333x + x2 = 0 + 1.611111111 Combine like terms: 0 + 1.611111111 = 1.611111111 2.333333333x + x2 = 1.611111111 The x term is 2.333333333x. Take half its coefficient (1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. 2.333333333x + 1.361111112 + x2 = 1.611111111 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333x + x2 = 1.611111111 + 1.361111112 Combine like terms: 1.611111111 + 1.361111112 = 2.972222223 1.361111112 + 2.333333333x + x2 = 2.972222223 Factor a perfect square on the left side: (x + 1.166666667)(x + 1.166666667) = 2.972222223 Calculate the square root of the right side: 1.724013406 Break this problem into two subproblems by setting (x + 1.166666667) equal to 1.724013406 and -1.724013406.Subproblem 1
x + 1.166666667 = 1.724013406 Simplifying x + 1.166666667 = 1.724013406 Reorder the terms: 1.166666667 + x = 1.724013406 Solving 1.166666667 + x = 1.724013406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = 1.724013406 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 1.724013406 + -1.166666667 x = 1.724013406 + -1.166666667 Combine like terms: 1.724013406 + -1.166666667 = 0.557346739 x = 0.557346739 Simplifying x = 0.557346739Subproblem 2
x + 1.166666667 = -1.724013406 Simplifying x + 1.166666667 = -1.724013406 Reorder the terms: 1.166666667 + x = -1.724013406 Solving 1.166666667 + x = -1.724013406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = -1.724013406 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -1.724013406 + -1.166666667 x = -1.724013406 + -1.166666667 Combine like terms: -1.724013406 + -1.166666667 = -2.890680073 x = -2.890680073 Simplifying x = -2.890680073Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.557346739, -2.890680073}
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