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Simplifying -0.013x2 + 0.25x + 1 = 0.5 Reorder the terms: 1 + 0.25x + -0.013x2 = 0.5 Solving 1 + 0.25x + -0.013x2 = 0.5 Solving for variable 'x'. Reorder the terms: 1 + -0.5 + 0.25x + -0.013x2 = 0.5 + -0.5 Combine like terms: 1 + -0.5 = 0.5 0.5 + 0.25x + -0.013x2 = 0.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.5 + 0.25x + -0.013x2 = 0.0 Begin completing the square. Divide all terms by -0.013 the coefficient of the squared term: Divide each side by '-0.013'. -38.46153846 + -19.23076923x + x2 = 0 Move the constant term to the right: Add '38.46153846' to each side of the equation. -38.46153846 + -19.23076923x + 38.46153846 + x2 = 0 + 38.46153846 Reorder the terms: -38.46153846 + 38.46153846 + -19.23076923x + x2 = 0 + 38.46153846 Combine like terms: -38.46153846 + 38.46153846 = 0.00000000 0.00000000 + -19.23076923x + x2 = 0 + 38.46153846 -19.23076923x + x2 = 0 + 38.46153846 Combine like terms: 0 + 38.46153846 = 38.46153846 -19.23076923x + x2 = 38.46153846 The x term is -19.23076923x. Take half its coefficient (-9.615384615). Square it (92.45562129) and add it to both sides. Add '92.45562129' to each side of the equation. -19.23076923x + 92.45562129 + x2 = 38.46153846 + 92.45562129 Reorder the terms: 92.45562129 + -19.23076923x + x2 = 38.46153846 + 92.45562129 Combine like terms: 38.46153846 + 92.45562129 = 130.91715975 92.45562129 + -19.23076923x + x2 = 130.91715975 Factor a perfect square on the left side: (x + -9.615384615)(x + -9.615384615) = 130.91715975 Calculate the square root of the right side: 11.441903677 Break this problem into two subproblems by setting (x + -9.615384615) equal to 11.441903677 and -11.441903677.Subproblem 1
x + -9.615384615 = 11.441903677 Simplifying x + -9.615384615 = 11.441903677 Reorder the terms: -9.615384615 + x = 11.441903677 Solving -9.615384615 + x = 11.441903677 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.615384615' to each side of the equation. -9.615384615 + 9.615384615 + x = 11.441903677 + 9.615384615 Combine like terms: -9.615384615 + 9.615384615 = 0.000000000 0.000000000 + x = 11.441903677 + 9.615384615 x = 11.441903677 + 9.615384615 Combine like terms: 11.441903677 + 9.615384615 = 21.057288292 x = 21.057288292 Simplifying x = 21.057288292Subproblem 2
x + -9.615384615 = -11.441903677 Simplifying x + -9.615384615 = -11.441903677 Reorder the terms: -9.615384615 + x = -11.441903677 Solving -9.615384615 + x = -11.441903677 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9.615384615' to each side of the equation. -9.615384615 + 9.615384615 + x = -11.441903677 + 9.615384615 Combine like terms: -9.615384615 + 9.615384615 = 0.000000000 0.000000000 + x = -11.441903677 + 9.615384615 x = -11.441903677 + 9.615384615 Combine like terms: -11.441903677 + 9.615384615 = -1.826519062 x = -1.826519062 Simplifying x = -1.826519062Solution
The solution to the problem is based on the solutions from the subproblems. x = {21.057288292, -1.826519062}
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