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Simplifying 9x2 + 23 = 30x Reorder the terms: 23 + 9x2 = 30x Solving 23 + 9x2 = 30x Solving for variable 'x'. Reorder the terms: 23 + -30x + 9x2 = 30x + -30x Combine like terms: 30x + -30x = 0 23 + -30x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 2.555555556 + -3.333333333x + x2 = 0 Move the constant term to the right: Add '-2.555555556' to each side of the equation. 2.555555556 + -3.333333333x + -2.555555556 + x2 = 0 + -2.555555556 Reorder the terms: 2.555555556 + -2.555555556 + -3.333333333x + x2 = 0 + -2.555555556 Combine like terms: 2.555555556 + -2.555555556 = 0.000000000 0.000000000 + -3.333333333x + x2 = 0 + -2.555555556 -3.333333333x + x2 = 0 + -2.555555556 Combine like terms: 0 + -2.555555556 = -2.555555556 -3.333333333x + x2 = -2.555555556 The x term is -3.333333333x. Take half its coefficient (-1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. -3.333333333x + 2.777777779 + x2 = -2.555555556 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333x + x2 = -2.555555556 + 2.777777779 Combine like terms: -2.555555556 + 2.777777779 = 0.222222223 2.777777779 + -3.333333333x + x2 = 0.222222223 Factor a perfect square on the left side: (x + -1.666666667)(x + -1.666666667) = 0.222222223 Calculate the square root of the right side: 0.471404522 Break this problem into two subproblems by setting (x + -1.666666667) equal to 0.471404522 and -0.471404522.Subproblem 1
x + -1.666666667 = 0.471404522 Simplifying x + -1.666666667 = 0.471404522 Reorder the terms: -1.666666667 + x = 0.471404522 Solving -1.666666667 + x = 0.471404522 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + x = 0.471404522 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + x = 0.471404522 + 1.666666667 x = 0.471404522 + 1.666666667 Combine like terms: 0.471404522 + 1.666666667 = 2.138071189 x = 2.138071189 Simplifying x = 2.138071189Subproblem 2
x + -1.666666667 = -0.471404522 Simplifying x + -1.666666667 = -0.471404522 Reorder the terms: -1.666666667 + x = -0.471404522 Solving -1.666666667 + x = -0.471404522 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + x = -0.471404522 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + x = -0.471404522 + 1.666666667 x = -0.471404522 + 1.666666667 Combine like terms: -0.471404522 + 1.666666667 = 1.195262145 x = 1.195262145 Simplifying x = 1.195262145Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.138071189, 1.195262145}
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