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Simplifying -1.5x2 + 1.5x + 45 = 0.1x2 + 0.5x + 10 Reorder the terms: 45 + 1.5x + -1.5x2 = 0.1x2 + 0.5x + 10 Reorder the terms: 45 + 1.5x + -1.5x2 = 10 + 0.5x + 0.1x2 Solving 45 + 1.5x + -1.5x2 = 10 + 0.5x + 0.1x2 Solving for variable 'x'. Reorder the terms: 45 + -10 + 1.5x + -0.5x + -1.5x2 + -0.1x2 = 10 + 0.5x + 0.1x2 + -10 + -0.5x + -0.1x2 Combine like terms: 45 + -10 = 35 35 + 1.5x + -0.5x + -1.5x2 + -0.1x2 = 10 + 0.5x + 0.1x2 + -10 + -0.5x + -0.1x2 Combine like terms: 1.5x + -0.5x = 1x 35 + 1x + -1.5x2 + -0.1x2 = 10 + 0.5x + 0.1x2 + -10 + -0.5x + -0.1x2 Combine like terms: -1.5x2 + -0.1x2 = -1.6x2 35 + 1x + -1.6x2 = 10 + 0.5x + 0.1x2 + -10 + -0.5x + -0.1x2 Reorder the terms: 35 + 1x + -1.6x2 = 10 + -10 + 0.5x + -0.5x + 0.1x2 + -0.1x2 Combine like terms: 10 + -10 = 0 35 + 1x + -1.6x2 = 0 + 0.5x + -0.5x + 0.1x2 + -0.1x2 35 + 1x + -1.6x2 = 0.5x + -0.5x + 0.1x2 + -0.1x2 Combine like terms: 0.5x + -0.5x = 0.0 35 + 1x + -1.6x2 = 0.0 + 0.1x2 + -0.1x2 35 + 1x + -1.6x2 = 0.1x2 + -0.1x2 Combine like terms: 0.1x2 + -0.1x2 = 0.0 35 + 1x + -1.6x2 = 0.0 Begin completing the square. Divide all terms by -1.6 the coefficient of the squared term: Divide each side by '-1.6'. -21.875 + -0.625x + x2 = 0 Move the constant term to the right: Add '21.875' to each side of the equation. -21.875 + -0.625x + 21.875 + x2 = 0 + 21.875 Reorder the terms: -21.875 + 21.875 + -0.625x + x2 = 0 + 21.875 Combine like terms: -21.875 + 21.875 = 0.000 0.000 + -0.625x + x2 = 0 + 21.875 -0.625x + x2 = 0 + 21.875 Combine like terms: 0 + 21.875 = 21.875 -0.625x + x2 = 21.875 The x term is -0.625x. Take half its coefficient (-0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. -0.625x + 0.09765625 + x2 = 21.875 + 0.09765625 Reorder the terms: 0.09765625 + -0.625x + x2 = 21.875 + 0.09765625 Combine like terms: 21.875 + 0.09765625 = 21.97265625 0.09765625 + -0.625x + x2 = 21.97265625 Factor a perfect square on the left side: (x + -0.3125)(x + -0.3125) = 21.97265625 Calculate the square root of the right side: 4.6875 Break this problem into two subproblems by setting (x + -0.3125) equal to 4.6875 and -4.6875.Subproblem 1
x + -0.3125 = 4.6875 Simplifying x + -0.3125 = 4.6875 Reorder the terms: -0.3125 + x = 4.6875 Solving -0.3125 + x = 4.6875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + x = 4.6875 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + x = 4.6875 + 0.3125 x = 4.6875 + 0.3125 Combine like terms: 4.6875 + 0.3125 = 5 x = 5 Simplifying x = 5Subproblem 2
x + -0.3125 = -4.6875 Simplifying x + -0.3125 = -4.6875 Reorder the terms: -0.3125 + x = -4.6875 Solving -0.3125 + x = -4.6875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + x = -4.6875 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + x = -4.6875 + 0.3125 x = -4.6875 + 0.3125 Combine like terms: -4.6875 + 0.3125 = -4.375 x = -4.375 Simplifying x = -4.375Solution
The solution to the problem is based on the solutions from the subproblems. x = {5, -4.375}
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