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Simplifying 9x2 + -24x = 1 Reorder the terms: -24x + 9x2 = 1 Solving -24x + 9x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + -24x + 9x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -24x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + -2.666666667x + x2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + -2.666666667x + 0.1111111111 + x2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + -2.666666667x + x2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + -2.666666667x + x2 = 0 + 0.1111111111 -2.666666667x + x2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 -2.666666667x + x2 = 0.1111111111 The x term is -2.666666667x. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667x + 1.777777780 + x2 = 0.1111111111 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667x + x2 = 0.1111111111 + 1.777777780 Combine like terms: 0.1111111111 + 1.777777780 = 1.8888888911 1.777777780 + -2.666666667x + x2 = 1.8888888911 Factor a perfect square on the left side: (x + -1.333333334)(x + -1.333333334) = 1.8888888911 Calculate the square root of the right side: 1.374368543 Break this problem into two subproblems by setting (x + -1.333333334) equal to 1.374368543 and -1.374368543.Subproblem 1
x + -1.333333334 = 1.374368543 Simplifying x + -1.333333334 = 1.374368543 Reorder the terms: -1.333333334 + x = 1.374368543 Solving -1.333333334 + x = 1.374368543 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = 1.374368543 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 1.374368543 + 1.333333334 x = 1.374368543 + 1.333333334 Combine like terms: 1.374368543 + 1.333333334 = 2.707701877 x = 2.707701877 Simplifying x = 2.707701877Subproblem 2
x + -1.333333334 = -1.374368543 Simplifying x + -1.333333334 = -1.374368543 Reorder the terms: -1.333333334 + x = -1.374368543 Solving -1.333333334 + x = -1.374368543 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = -1.374368543 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -1.374368543 + 1.333333334 x = -1.374368543 + 1.333333334 Combine like terms: -1.374368543 + 1.333333334 = -0.041035209 x = -0.041035209 Simplifying x = -0.041035209Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.707701877, -0.041035209}
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