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Simplifying 70 = 0.14x2 + 0.65x + 7.69 Reorder the terms: 70 = 7.69 + 0.65x + 0.14x2 Solving 70 = 7.69 + 0.65x + 0.14x2 Solving for variable 'x'. Combine like terms: 70 + -7.69 = 62.31 62.31 + -0.65x + -0.14x2 = 7.69 + 0.65x + 0.14x2 + -7.69 + -0.65x + -0.14x2 Reorder the terms: 62.31 + -0.65x + -0.14x2 = 7.69 + -7.69 + 0.65x + -0.65x + 0.14x2 + -0.14x2 Combine like terms: 7.69 + -7.69 = 0.00 62.31 + -0.65x + -0.14x2 = 0.00 + 0.65x + -0.65x + 0.14x2 + -0.14x2 62.31 + -0.65x + -0.14x2 = 0.65x + -0.65x + 0.14x2 + -0.14x2 Combine like terms: 0.65x + -0.65x = 0.00 62.31 + -0.65x + -0.14x2 = 0.00 + 0.14x2 + -0.14x2 62.31 + -0.65x + -0.14x2 = 0.14x2 + -0.14x2 Combine like terms: 0.14x2 + -0.14x2 = 0.00 62.31 + -0.65x + -0.14x2 = 0.00 Begin completing the square. Divide all terms by -0.14 the coefficient of the squared term: Divide each side by '-0.14'. -445.0714286 + 4.642857143x + x2 = 0 Move the constant term to the right: Add '445.0714286' to each side of the equation. -445.0714286 + 4.642857143x + 445.0714286 + x2 = 0 + 445.0714286 Reorder the terms: -445.0714286 + 445.0714286 + 4.642857143x + x2 = 0 + 445.0714286 Combine like terms: -445.0714286 + 445.0714286 = 0.0000000 0.0000000 + 4.642857143x + x2 = 0 + 445.0714286 4.642857143x + x2 = 0 + 445.0714286 Combine like terms: 0 + 445.0714286 = 445.0714286 4.642857143x + x2 = 445.0714286 The x term is 4.642857143x. Take half its coefficient (2.321428572). Square it (5.389030615) and add it to both sides. Add '5.389030615' to each side of the equation. 4.642857143x + 5.389030615 + x2 = 445.0714286 + 5.389030615 Reorder the terms: 5.389030615 + 4.642857143x + x2 = 445.0714286 + 5.389030615 Combine like terms: 445.0714286 + 5.389030615 = 450.460459215 5.389030615 + 4.642857143x + x2 = 450.460459215 Factor a perfect square on the left side: (x + 2.321428572)(x + 2.321428572) = 450.460459215 Calculate the square root of the right side: 21.224053788 Break this problem into two subproblems by setting (x + 2.321428572) equal to 21.224053788 and -21.224053788.Subproblem 1
x + 2.321428572 = 21.224053788 Simplifying x + 2.321428572 = 21.224053788 Reorder the terms: 2.321428572 + x = 21.224053788 Solving 2.321428572 + x = 21.224053788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.321428572' to each side of the equation. 2.321428572 + -2.321428572 + x = 21.224053788 + -2.321428572 Combine like terms: 2.321428572 + -2.321428572 = 0.000000000 0.000000000 + x = 21.224053788 + -2.321428572 x = 21.224053788 + -2.321428572 Combine like terms: 21.224053788 + -2.321428572 = 18.902625216 x = 18.902625216 Simplifying x = 18.902625216Subproblem 2
x + 2.321428572 = -21.224053788 Simplifying x + 2.321428572 = -21.224053788 Reorder the terms: 2.321428572 + x = -21.224053788 Solving 2.321428572 + x = -21.224053788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.321428572' to each side of the equation. 2.321428572 + -2.321428572 + x = -21.224053788 + -2.321428572 Combine like terms: 2.321428572 + -2.321428572 = 0.000000000 0.000000000 + x = -21.224053788 + -2.321428572 x = -21.224053788 + -2.321428572 Combine like terms: -21.224053788 + -2.321428572 = -23.54548236 x = -23.54548236 Simplifying x = -23.54548236Solution
The solution to the problem is based on the solutions from the subproblems. x = {18.902625216, -23.54548236}
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