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Simplifying 1.2 + 0.8x = 0.75x(x + 6) Reorder the terms: 1.2 + 0.8x = 0.75x(6 + x) 1.2 + 0.8x = (6 * 0.75x + x * 0.75x) 1.2 + 0.8x = (4.5x + 0.75x2) Solving 1.2 + 0.8x = 4.5x + 0.75x2 Solving for variable 'x'. Combine like terms: 0.8x + -4.5x = -3.7x 1.2 + -3.7x + -0.75x2 = 4.5x + 0.75x2 + -4.5x + -0.75x2 Reorder the terms: 1.2 + -3.7x + -0.75x2 = 4.5x + -4.5x + 0.75x2 + -0.75x2 Combine like terms: 4.5x + -4.5x = 0.0 1.2 + -3.7x + -0.75x2 = 0.0 + 0.75x2 + -0.75x2 1.2 + -3.7x + -0.75x2 = 0.75x2 + -0.75x2 Combine like terms: 0.75x2 + -0.75x2 = 0.00 1.2 + -3.7x + -0.75x2 = 0.00 Begin completing the square. Divide all terms by -0.75 the coefficient of the squared term: Divide each side by '-0.75'. -1.6 + 4.933333333x + x2 = 0 Move the constant term to the right: Add '1.6' to each side of the equation. -1.6 + 4.933333333x + 1.6 + x2 = 0 + 1.6 Reorder the terms: -1.6 + 1.6 + 4.933333333x + x2 = 0 + 1.6 Combine like terms: -1.6 + 1.6 = 0.0 0.0 + 4.933333333x + x2 = 0 + 1.6 4.933333333x + x2 = 0 + 1.6 Combine like terms: 0 + 1.6 = 1.6 4.933333333x + x2 = 1.6 The x term is 4.933333333x. Take half its coefficient (2.466666667). Square it (6.084444446) and add it to both sides. Add '6.084444446' to each side of the equation. 4.933333333x + 6.084444446 + x2 = 1.6 + 6.084444446 Reorder the terms: 6.084444446 + 4.933333333x + x2 = 1.6 + 6.084444446 Combine like terms: 1.6 + 6.084444446 = 7.684444446 6.084444446 + 4.933333333x + x2 = 7.684444446 Factor a perfect square on the left side: (x + 2.466666667)(x + 2.466666667) = 7.684444446 Calculate the square root of the right side: 2.772083052 Break this problem into two subproblems by setting (x + 2.466666667) equal to 2.772083052 and -2.772083052.Subproblem 1
x + 2.466666667 = 2.772083052 Simplifying x + 2.466666667 = 2.772083052 Reorder the terms: 2.466666667 + x = 2.772083052 Solving 2.466666667 + x = 2.772083052 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.466666667' to each side of the equation. 2.466666667 + -2.466666667 + x = 2.772083052 + -2.466666667 Combine like terms: 2.466666667 + -2.466666667 = 0.000000000 0.000000000 + x = 2.772083052 + -2.466666667 x = 2.772083052 + -2.466666667 Combine like terms: 2.772083052 + -2.466666667 = 0.305416385 x = 0.305416385 Simplifying x = 0.305416385Subproblem 2
x + 2.466666667 = -2.772083052 Simplifying x + 2.466666667 = -2.772083052 Reorder the terms: 2.466666667 + x = -2.772083052 Solving 2.466666667 + x = -2.772083052 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.466666667' to each side of the equation. 2.466666667 + -2.466666667 + x = -2.772083052 + -2.466666667 Combine like terms: 2.466666667 + -2.466666667 = 0.000000000 0.000000000 + x = -2.772083052 + -2.466666667 x = -2.772083052 + -2.466666667 Combine like terms: -2.772083052 + -2.466666667 = -5.238749719 x = -5.238749719 Simplifying x = -5.238749719Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.305416385, -5.238749719}
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