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Simplifying 105 = 1.5x2 + 10.5x + 15 Reorder the terms: 105 = 15 + 10.5x + 1.5x2 Solving 105 = 15 + 10.5x + 1.5x2 Solving for variable 'x'. Combine like terms: 105 + -15 = 90 90 + -10.5x + -1.5x2 = 15 + 10.5x + 1.5x2 + -15 + -10.5x + -1.5x2 Reorder the terms: 90 + -10.5x + -1.5x2 = 15 + -15 + 10.5x + -10.5x + 1.5x2 + -1.5x2 Combine like terms: 15 + -15 = 0 90 + -10.5x + -1.5x2 = 0 + 10.5x + -10.5x + 1.5x2 + -1.5x2 90 + -10.5x + -1.5x2 = 10.5x + -10.5x + 1.5x2 + -1.5x2 Combine like terms: 10.5x + -10.5x = 0.0 90 + -10.5x + -1.5x2 = 0.0 + 1.5x2 + -1.5x2 90 + -10.5x + -1.5x2 = 1.5x2 + -1.5x2 Combine like terms: 1.5x2 + -1.5x2 = 0.0 90 + -10.5x + -1.5x2 = 0.0 Begin completing the square. Divide all terms by -1.5 the coefficient of the squared term: Divide each side by '-1.5'. -60 + 7x + x2 = 0 Move the constant term to the right: Add '60' to each side of the equation. -60 + 7x + 60 + x2 = 0 + 60 Reorder the terms: -60 + 60 + 7x + x2 = 0 + 60 Combine like terms: -60 + 60 = 0 0 + 7x + x2 = 0 + 60 7x + x2 = 0 + 60 Combine like terms: 0 + 60 = 60 7x + x2 = 60 The x term is 7x. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7x + 12.25 + x2 = 60 + 12.25 Reorder the terms: 12.25 + 7x + x2 = 60 + 12.25 Combine like terms: 60 + 12.25 = 72.25 12.25 + 7x + x2 = 72.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 72.25 Calculate the square root of the right side: 8.5 Break this problem into two subproblems by setting (x + 3.5) equal to 8.5 and -8.5.Subproblem 1
x + 3.5 = 8.5 Simplifying x + 3.5 = 8.5 Reorder the terms: 3.5 + x = 8.5 Solving 3.5 + x = 8.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 8.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 8.5 + -3.5 x = 8.5 + -3.5 Combine like terms: 8.5 + -3.5 = 5 x = 5 Simplifying x = 5Subproblem 2
x + 3.5 = -8.5 Simplifying x + 3.5 = -8.5 Reorder the terms: 3.5 + x = -8.5 Solving 3.5 + x = -8.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -8.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -8.5 + -3.5 x = -8.5 + -3.5 Combine like terms: -8.5 + -3.5 = -12 x = -12 Simplifying x = -12Solution
The solution to the problem is based on the solutions from the subproblems. x = {5, -12}
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