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Simplifying 9(2x + -1)(2x + -1) + -8 = 0 Reorder the terms: 9(-1 + 2x)(2x + -1) + -8 = 0 Reorder the terms: 9(-1 + 2x)(-1 + 2x) + -8 = 0 Multiply (-1 + 2x) * (-1 + 2x) 9(-1(-1 + 2x) + 2x * (-1 + 2x)) + -8 = 0 9((-1 * -1 + 2x * -1) + 2x * (-1 + 2x)) + -8 = 0 9((1 + -2x) + 2x * (-1 + 2x)) + -8 = 0 9(1 + -2x + (-1 * 2x + 2x * 2x)) + -8 = 0 9(1 + -2x + (-2x + 4x2)) + -8 = 0 Combine like terms: -2x + -2x = -4x 9(1 + -4x + 4x2) + -8 = 0 (1 * 9 + -4x * 9 + 4x2 * 9) + -8 = 0 (9 + -36x + 36x2) + -8 = 0 Reorder the terms: 9 + -8 + -36x + 36x2 = 0 Combine like terms: 9 + -8 = 1 1 + -36x + 36x2 = 0 Solving 1 + -36x + 36x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 36 the coefficient of the squared term: Divide each side by '36'. 0.02777777778 + -1x + x2 = 0 Move the constant term to the right: Add '-0.02777777778' to each side of the equation. 0.02777777778 + -1x + -0.02777777778 + x2 = 0 + -0.02777777778 Reorder the terms: 0.02777777778 + -0.02777777778 + -1x + x2 = 0 + -0.02777777778 Combine like terms: 0.02777777778 + -0.02777777778 = 0.00000000000 0.00000000000 + -1x + x2 = 0 + -0.02777777778 -1x + x2 = 0 + -0.02777777778 Combine like terms: 0 + -0.02777777778 = -0.02777777778 -1x + x2 = -0.02777777778 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = -0.02777777778 + 0.25 Reorder the terms: 0.25 + -1x + x2 = -0.02777777778 + 0.25 Combine like terms: -0.02777777778 + 0.25 = 0.22222222222 0.25 + -1x + x2 = 0.22222222222 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.22222222222 Calculate the square root of the right side: 0.471404521 Break this problem into two subproblems by setting (x + -0.5) equal to 0.471404521 and -0.471404521.Subproblem 1
x + -0.5 = 0.471404521 Simplifying x + -0.5 = 0.471404521 Reorder the terms: -0.5 + x = 0.471404521 Solving -0.5 + x = 0.471404521 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 0.471404521 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.471404521 + 0.5 x = 0.471404521 + 0.5 Combine like terms: 0.471404521 + 0.5 = 0.971404521 x = 0.971404521 Simplifying x = 0.971404521Subproblem 2
x + -0.5 = -0.471404521 Simplifying x + -0.5 = -0.471404521 Reorder the terms: -0.5 + x = -0.471404521 Solving -0.5 + x = -0.471404521 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -0.471404521 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.471404521 + 0.5 x = -0.471404521 + 0.5 Combine like terms: -0.471404521 + 0.5 = 0.028595479 x = 0.028595479 Simplifying x = 0.028595479Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.971404521, 0.028595479}
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