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Simplifying 7 = -0.1x2 + -0.3x + 11 Reorder the terms: 7 = 11 + -0.3x + -0.1x2 Solving 7 = 11 + -0.3x + -0.1x2 Solving for variable 'x'. Combine like terms: 7 + -11 = -4 -4 + 0.3x + 0.1x2 = 11 + -0.3x + -0.1x2 + -11 + 0.3x + 0.1x2 Reorder the terms: -4 + 0.3x + 0.1x2 = 11 + -11 + -0.3x + 0.3x + -0.1x2 + 0.1x2 Combine like terms: 11 + -11 = 0 -4 + 0.3x + 0.1x2 = 0 + -0.3x + 0.3x + -0.1x2 + 0.1x2 -4 + 0.3x + 0.1x2 = -0.3x + 0.3x + -0.1x2 + 0.1x2 Combine like terms: -0.3x + 0.3x = 0.0 -4 + 0.3x + 0.1x2 = 0.0 + -0.1x2 + 0.1x2 -4 + 0.3x + 0.1x2 = -0.1x2 + 0.1x2 Combine like terms: -0.1x2 + 0.1x2 = 0.0 -4 + 0.3x + 0.1x2 = 0.0 Begin completing the square. Divide all terms by 0.1 the coefficient of the squared term: Divide each side by '0.1'. -40 + 3x + x2 = 0 Move the constant term to the right: Add '40' to each side of the equation. -40 + 3x + 40 + x2 = 0 + 40 Reorder the terms: -40 + 40 + 3x + x2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + 3x + x2 = 0 + 40 3x + x2 = 0 + 40 Combine like terms: 0 + 40 = 40 3x + x2 = 40 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = 40 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 40 + 2.25 Combine like terms: 40 + 2.25 = 42.25 2.25 + 3x + x2 = 42.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 42.25 Calculate the square root of the right side: 6.5 Break this problem into two subproblems by setting (x + 1.5) equal to 6.5 and -6.5.Subproblem 1
x + 1.5 = 6.5 Simplifying x + 1.5 = 6.5 Reorder the terms: 1.5 + x = 6.5 Solving 1.5 + x = 6.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 6.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 6.5 + -1.5 x = 6.5 + -1.5 Combine like terms: 6.5 + -1.5 = 5 x = 5 Simplifying x = 5Subproblem 2
x + 1.5 = -6.5 Simplifying x + 1.5 = -6.5 Reorder the terms: 1.5 + x = -6.5 Solving 1.5 + x = -6.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -6.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -6.5 + -1.5 x = -6.5 + -1.5 Combine like terms: -6.5 + -1.5 = -8 x = -8 Simplifying x = -8Solution
The solution to the problem is based on the solutions from the subproblems. x = {5, -8}
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