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Simplifying 4.5(x + -2) * 1.5x = (3x + -4) + -1 Reorder the terms: 4.5(-2 + x) * 1.5x = (3x + -4) + -1 Reorder the terms for easier multiplication: 4.5 * 1.5x(-2 + x) = (3x + -4) + -1 Multiply 4.5 * 1.5 6.75x(-2 + x) = (3x + -4) + -1 (-2 * 6.75x + x * 6.75x) = (3x + -4) + -1 (-13.5x + 6.75x2) = (3x + -4) + -1 Reorder the terms: -13.5x + 6.75x2 = (-4 + 3x) + -1 Remove parenthesis around (-4 + 3x) -13.5x + 6.75x2 = -4 + 3x + -1 Reorder the terms: -13.5x + 6.75x2 = -4 + -1 + 3x Combine like terms: -4 + -1 = -5 -13.5x + 6.75x2 = -5 + 3x Solving -13.5x + 6.75x2 = -5 + 3x Solving for variable 'x'. Reorder the terms: 5 + -13.5x + -3x + 6.75x2 = -5 + 3x + 5 + -3x Combine like terms: -13.5x + -3x = -16.5x 5 + -16.5x + 6.75x2 = -5 + 3x + 5 + -3x Reorder the terms: 5 + -16.5x + 6.75x2 = -5 + 5 + 3x + -3x Combine like terms: -5 + 5 = 0 5 + -16.5x + 6.75x2 = 0 + 3x + -3x 5 + -16.5x + 6.75x2 = 3x + -3x Combine like terms: 3x + -3x = 0 5 + -16.5x + 6.75x2 = 0 Begin completing the square. Divide all terms by 6.75 the coefficient of the squared term: Divide each side by '6.75'. 0.7407407407 + -2.444444444x + x2 = 0 Move the constant term to the right: Add '-0.7407407407' to each side of the equation. 0.7407407407 + -2.444444444x + -0.7407407407 + x2 = 0 + -0.7407407407 Reorder the terms: 0.7407407407 + -0.7407407407 + -2.444444444x + x2 = 0 + -0.7407407407 Combine like terms: 0.7407407407 + -0.7407407407 = 0.0000000000 0.0000000000 + -2.444444444x + x2 = 0 + -0.7407407407 -2.444444444x + x2 = 0 + -0.7407407407 Combine like terms: 0 + -0.7407407407 = -0.7407407407 -2.444444444x + x2 = -0.7407407407 The x term is -2.444444444x. Take half its coefficient (-1.222222222). Square it (1.493827160) and add it to both sides. Add '1.493827160' to each side of the equation. -2.444444444x + 1.493827160 + x2 = -0.7407407407 + 1.493827160 Reorder the terms: 1.493827160 + -2.444444444x + x2 = -0.7407407407 + 1.493827160 Combine like terms: -0.7407407407 + 1.493827160 = 0.7530864193 1.493827160 + -2.444444444x + x2 = 0.7530864193 Factor a perfect square on the left side: (x + -1.222222222)(x + -1.222222222) = 0.7530864193 Calculate the square root of the right side: 0.867805519 Break this problem into two subproblems by setting (x + -1.222222222) equal to 0.867805519 and -0.867805519.Subproblem 1
x + -1.222222222 = 0.867805519 Simplifying x + -1.222222222 = 0.867805519 Reorder the terms: -1.222222222 + x = 0.867805519 Solving -1.222222222 + x = 0.867805519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.222222222' to each side of the equation. -1.222222222 + 1.222222222 + x = 0.867805519 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + x = 0.867805519 + 1.222222222 x = 0.867805519 + 1.222222222 Combine like terms: 0.867805519 + 1.222222222 = 2.090027741 x = 2.090027741 Simplifying x = 2.090027741Subproblem 2
x + -1.222222222 = -0.867805519 Simplifying x + -1.222222222 = -0.867805519 Reorder the terms: -1.222222222 + x = -0.867805519 Solving -1.222222222 + x = -0.867805519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.222222222' to each side of the equation. -1.222222222 + 1.222222222 + x = -0.867805519 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + x = -0.867805519 + 1.222222222 x = -0.867805519 + 1.222222222 Combine like terms: -0.867805519 + 1.222222222 = 0.354416703 x = 0.354416703 Simplifying x = 0.354416703Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.090027741, 0.354416703}
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