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Simplifying 8 = -0.021x2 + 0.6x + 6 Reorder the terms: 8 = 6 + 0.6x + -0.021x2 Solving 8 = 6 + 0.6x + -0.021x2 Solving for variable 'x'. Combine like terms: 8 + -6 = 2 2 + -0.6x + 0.021x2 = 6 + 0.6x + -0.021x2 + -6 + -0.6x + 0.021x2 Reorder the terms: 2 + -0.6x + 0.021x2 = 6 + -6 + 0.6x + -0.6x + -0.021x2 + 0.021x2 Combine like terms: 6 + -6 = 0 2 + -0.6x + 0.021x2 = 0 + 0.6x + -0.6x + -0.021x2 + 0.021x2 2 + -0.6x + 0.021x2 = 0.6x + -0.6x + -0.021x2 + 0.021x2 Combine like terms: 0.6x + -0.6x = 0.0 2 + -0.6x + 0.021x2 = 0.0 + -0.021x2 + 0.021x2 2 + -0.6x + 0.021x2 = -0.021x2 + 0.021x2 Combine like terms: -0.021x2 + 0.021x2 = 0.000 2 + -0.6x + 0.021x2 = 0.000 Begin completing the square. Divide all terms by 0.021 the coefficient of the squared term: Divide each side by '0.021'. 95.23809524 + -28.57142857x + x2 = 0 Move the constant term to the right: Add '-95.23809524' to each side of the equation. 95.23809524 + -28.57142857x + -95.23809524 + x2 = 0 + -95.23809524 Reorder the terms: 95.23809524 + -95.23809524 + -28.57142857x + x2 = 0 + -95.23809524 Combine like terms: 95.23809524 + -95.23809524 = 0.00000000 0.00000000 + -28.57142857x + x2 = 0 + -95.23809524 -28.57142857x + x2 = 0 + -95.23809524 Combine like terms: 0 + -95.23809524 = -95.23809524 -28.57142857x + x2 = -95.23809524 The x term is -28.57142857x. Take half its coefficient (-14.28571429). Square it (204.0816328) and add it to both sides. Add '204.0816328' to each side of the equation. -28.57142857x + 204.0816328 + x2 = -95.23809524 + 204.0816328 Reorder the terms: 204.0816328 + -28.57142857x + x2 = -95.23809524 + 204.0816328 Combine like terms: -95.23809524 + 204.0816328 = 108.84353756 204.0816328 + -28.57142857x + x2 = 108.84353756 Factor a perfect square on the left side: (x + -14.28571429)(x + -14.28571429) = 108.84353756 Calculate the square root of the right side: 10.432810626 Break this problem into two subproblems by setting (x + -14.28571429) equal to 10.432810626 and -10.432810626.Subproblem 1
x + -14.28571429 = 10.432810626 Simplifying x + -14.28571429 = 10.432810626 Reorder the terms: -14.28571429 + x = 10.432810626 Solving -14.28571429 + x = 10.432810626 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '14.28571429' to each side of the equation. -14.28571429 + 14.28571429 + x = 10.432810626 + 14.28571429 Combine like terms: -14.28571429 + 14.28571429 = 0.00000000 0.00000000 + x = 10.432810626 + 14.28571429 x = 10.432810626 + 14.28571429 Combine like terms: 10.432810626 + 14.28571429 = 24.718524916 x = 24.718524916 Simplifying x = 24.718524916Subproblem 2
x + -14.28571429 = -10.432810626 Simplifying x + -14.28571429 = -10.432810626 Reorder the terms: -14.28571429 + x = -10.432810626 Solving -14.28571429 + x = -10.432810626 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '14.28571429' to each side of the equation. -14.28571429 + 14.28571429 + x = -10.432810626 + 14.28571429 Combine like terms: -14.28571429 + 14.28571429 = 0.00000000 0.00000000 + x = -10.432810626 + 14.28571429 x = -10.432810626 + 14.28571429 Combine like terms: -10.432810626 + 14.28571429 = 3.852903664 x = 3.852903664 Simplifying x = 3.852903664Solution
The solution to the problem is based on the solutions from the subproblems. x = {24.718524916, 3.852903664}
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