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Simplifying 0.2x(x + 1) + 0.5x = -0.3(x + -4) Reorder the terms: 0.2x(1 + x) + 0.5x = -0.3(x + -4) (1 * 0.2x + x * 0.2x) + 0.5x = -0.3(x + -4) (0.2x + 0.2x2) + 0.5x = -0.3(x + -4) Reorder the terms: 0.2x + 0.5x + 0.2x2 = -0.3(x + -4) Combine like terms: 0.2x + 0.5x = 0.7x 0.7x + 0.2x2 = -0.3(x + -4) Reorder the terms: 0.7x + 0.2x2 = -0.3(-4 + x) 0.7x + 0.2x2 = (-4 * -0.3 + x * -0.3) 0.7x + 0.2x2 = (1.2 + -0.3x) Solving 0.7x + 0.2x2 = 1.2 + -0.3x Solving for variable 'x'. Reorder the terms: -1.2 + 0.7x + 0.3x + 0.2x2 = 1.2 + -0.3x + -1.2 + 0.3x Combine like terms: 0.7x + 0.3x = 1x -1.2 + 1x + 0.2x2 = 1.2 + -0.3x + -1.2 + 0.3x Reorder the terms: -1.2 + 1x + 0.2x2 = 1.2 + -1.2 + -0.3x + 0.3x Combine like terms: 1.2 + -1.2 = 0.0 -1.2 + 1x + 0.2x2 = 0.0 + -0.3x + 0.3x -1.2 + 1x + 0.2x2 = -0.3x + 0.3x Combine like terms: -0.3x + 0.3x = 0.0 -1.2 + 1x + 0.2x2 = 0.0 Begin completing the square. Divide all terms by 0.2 the coefficient of the squared term: Divide each side by '0.2'. -6 + 5x + x2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 5x + 6 + x2 = 0 + 6 Reorder the terms: -6 + 6 + 5x + x2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 5x + x2 = 0 + 6 5x + x2 = 0 + 6 Combine like terms: 0 + 6 = 6 5x + x2 = 6 The x term is 5x. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5x + 6.25 + x2 = 6 + 6.25 Reorder the terms: 6.25 + 5x + x2 = 6 + 6.25 Combine like terms: 6 + 6.25 = 12.25 6.25 + 5x + x2 = 12.25 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 12.25 Calculate the square root of the right side: 3.5 Break this problem into two subproblems by setting (x + 2.5) equal to 3.5 and -3.5.Subproblem 1
x + 2.5 = 3.5 Simplifying x + 2.5 = 3.5 Reorder the terms: 2.5 + x = 3.5 Solving 2.5 + x = 3.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 3.5 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 3.5 + -2.5 x = 3.5 + -2.5 Combine like terms: 3.5 + -2.5 = 1 x = 1 Simplifying x = 1Subproblem 2
x + 2.5 = -3.5 Simplifying x + 2.5 = -3.5 Reorder the terms: 2.5 + x = -3.5 Solving 2.5 + x = -3.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -3.5 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -3.5 + -2.5 x = -3.5 + -2.5 Combine like terms: -3.5 + -2.5 = -6 x = -6 Simplifying x = -6Solution
The solution to the problem is based on the solutions from the subproblems. x = {1, -6}
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