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Simplifying 9x2 + 3x = 19 Reorder the terms: 3x + 9x2 = 19 Solving 3x + 9x2 = 19 Solving for variable 'x'. Reorder the terms: -19 + 3x + 9x2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + 3x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.111111111 + 0.3333333333x + x2 = 0 Move the constant term to the right: Add '2.111111111' to each side of the equation. -2.111111111 + 0.3333333333x + 2.111111111 + x2 = 0 + 2.111111111 Reorder the terms: -2.111111111 + 2.111111111 + 0.3333333333x + x2 = 0 + 2.111111111 Combine like terms: -2.111111111 + 2.111111111 = 0.000000000 0.000000000 + 0.3333333333x + x2 = 0 + 2.111111111 0.3333333333x + x2 = 0 + 2.111111111 Combine like terms: 0 + 2.111111111 = 2.111111111 0.3333333333x + x2 = 2.111111111 The x term is 0.3333333333x. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333x + 0.02777777779 + x2 = 2.111111111 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333x + x2 = 2.111111111 + 0.02777777779 Combine like terms: 2.111111111 + 0.02777777779 = 2.13888888879 0.02777777779 + 0.3333333333x + x2 = 2.13888888879 Factor a perfect square on the left side: (x + 0.1666666667)(x + 0.1666666667) = 2.13888888879 Calculate the square root of the right side: 1.462494065 Break this problem into two subproblems by setting (x + 0.1666666667) equal to 1.462494065 and -1.462494065.Subproblem 1
x + 0.1666666667 = 1.462494065 Simplifying x + 0.1666666667 = 1.462494065 Reorder the terms: 0.1666666667 + x = 1.462494065 Solving 0.1666666667 + x = 1.462494065 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = 1.462494065 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 1.462494065 + -0.1666666667 x = 1.462494065 + -0.1666666667 Combine like terms: 1.462494065 + -0.1666666667 = 1.2958273983 x = 1.2958273983 Simplifying x = 1.2958273983Subproblem 2
x + 0.1666666667 = -1.462494065 Simplifying x + 0.1666666667 = -1.462494065 Reorder the terms: 0.1666666667 + x = -1.462494065 Solving 0.1666666667 + x = -1.462494065 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = -1.462494065 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -1.462494065 + -0.1666666667 x = -1.462494065 + -0.1666666667 Combine like terms: -1.462494065 + -0.1666666667 = -1.6291607317 x = -1.6291607317 Simplifying x = -1.6291607317Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.2958273983, -1.6291607317}
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