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Simplifying 6.5 = -0.05x2 + 4.5x + -6.5 Reorder the terms: 6.5 = -6.5 + 4.5x + -0.05x2 Solving 6.5 = -6.5 + 4.5x + -0.05x2 Solving for variable 'x'. Combine like terms: 6.5 + 6.5 = 13 13 + -4.5x + 0.05x2 = -6.5 + 4.5x + -0.05x2 + 6.5 + -4.5x + 0.05x2 Reorder the terms: 13 + -4.5x + 0.05x2 = -6.5 + 6.5 + 4.5x + -4.5x + -0.05x2 + 0.05x2 Combine like terms: -6.5 + 6.5 = 0.0 13 + -4.5x + 0.05x2 = 0.0 + 4.5x + -4.5x + -0.05x2 + 0.05x2 13 + -4.5x + 0.05x2 = 4.5x + -4.5x + -0.05x2 + 0.05x2 Combine like terms: 4.5x + -4.5x = 0.0 13 + -4.5x + 0.05x2 = 0.0 + -0.05x2 + 0.05x2 13 + -4.5x + 0.05x2 = -0.05x2 + 0.05x2 Combine like terms: -0.05x2 + 0.05x2 = 0.00 13 + -4.5x + 0.05x2 = 0.00 Begin completing the square. Divide all terms by 0.05 the coefficient of the squared term: Divide each side by '0.05'. 260 + -90x + x2 = 0 Move the constant term to the right: Add '-260' to each side of the equation. 260 + -90x + -260 + x2 = 0 + -260 Reorder the terms: 260 + -260 + -90x + x2 = 0 + -260 Combine like terms: 260 + -260 = 0 0 + -90x + x2 = 0 + -260 -90x + x2 = 0 + -260 Combine like terms: 0 + -260 = -260 -90x + x2 = -260 The x term is -90x. Take half its coefficient (-45). Square it (2025) and add it to both sides. Add '2025' to each side of the equation. -90x + 2025 + x2 = -260 + 2025 Reorder the terms: 2025 + -90x + x2 = -260 + 2025 Combine like terms: -260 + 2025 = 1765 2025 + -90x + x2 = 1765 Factor a perfect square on the left side: (x + -45)(x + -45) = 1765 Calculate the square root of the right side: 42.011903075 Break this problem into two subproblems by setting (x + -45) equal to 42.011903075 and -42.011903075.Subproblem 1
x + -45 = 42.011903075 Simplifying x + -45 = 42.011903075 Reorder the terms: -45 + x = 42.011903075 Solving -45 + x = 42.011903075 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + x = 42.011903075 + 45 Combine like terms: -45 + 45 = 0 0 + x = 42.011903075 + 45 x = 42.011903075 + 45 Combine like terms: 42.011903075 + 45 = 87.011903075 x = 87.011903075 Simplifying x = 87.011903075Subproblem 2
x + -45 = -42.011903075 Simplifying x + -45 = -42.011903075 Reorder the terms: -45 + x = -42.011903075 Solving -45 + x = -42.011903075 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '45' to each side of the equation. -45 + 45 + x = -42.011903075 + 45 Combine like terms: -45 + 45 = 0 0 + x = -42.011903075 + 45 x = -42.011903075 + 45 Combine like terms: -42.011903075 + 45 = 2.988096925 x = 2.988096925 Simplifying x = 2.988096925Solution
The solution to the problem is based on the solutions from the subproblems. x = {87.011903075, 2.988096925}
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