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Simplifying 10X2 + 9X = -1 Reorder the terms: 9X + 10X2 = -1 Solving 9X + 10X2 = -1 Solving for variable 'X'. Reorder the terms: 1 + 9X + 10X2 = -1 + 1 Combine like terms: -1 + 1 = 0 1 + 9X + 10X2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 0.1 + 0.9X + X2 = 0 Move the constant term to the right: Add '-0.1' to each side of the equation. 0.1 + 0.9X + -0.1 + X2 = 0 + -0.1 Reorder the terms: 0.1 + -0.1 + 0.9X + X2 = 0 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + 0.9X + X2 = 0 + -0.1 0.9X + X2 = 0 + -0.1 Combine like terms: 0 + -0.1 = -0.1 0.9X + X2 = -0.1 The X term is 0.9X. Take half its coefficient (0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. 0.9X + 0.2025 + X2 = -0.1 + 0.2025 Reorder the terms: 0.2025 + 0.9X + X2 = -0.1 + 0.2025 Combine like terms: -0.1 + 0.2025 = 0.1025 0.2025 + 0.9X + X2 = 0.1025 Factor a perfect square on the left side: (X + 0.45)(X + 0.45) = 0.1025 Calculate the square root of the right side: 0.320156212 Break this problem into two subproblems by setting (X + 0.45) equal to 0.320156212 and -0.320156212.Subproblem 1
X + 0.45 = 0.320156212 Simplifying X + 0.45 = 0.320156212 Reorder the terms: 0.45 + X = 0.320156212 Solving 0.45 + X = 0.320156212 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + X = 0.320156212 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + X = 0.320156212 + -0.45 X = 0.320156212 + -0.45 Combine like terms: 0.320156212 + -0.45 = -0.129843788 X = -0.129843788 Simplifying X = -0.129843788Subproblem 2
X + 0.45 = -0.320156212 Simplifying X + 0.45 = -0.320156212 Reorder the terms: 0.45 + X = -0.320156212 Solving 0.45 + X = -0.320156212 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + X = -0.320156212 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + X = -0.320156212 + -0.45 X = -0.320156212 + -0.45 Combine like terms: -0.320156212 + -0.45 = -0.770156212 X = -0.770156212 Simplifying X = -0.770156212Solution
The solution to the problem is based on the solutions from the subproblems. X = {-0.129843788, -0.770156212}
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