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Simplifying 10 = -5x2 + 15x + 0.5 Reorder the terms: 10 = 0.5 + 15x + -5x2 Solving 10 = 0.5 + 15x + -5x2 Solving for variable 'x'. Combine like terms: 10 + -0.5 = 9.5 9.5 + -15x + 5x2 = 0.5 + 15x + -5x2 + -0.5 + -15x + 5x2 Reorder the terms: 9.5 + -15x + 5x2 = 0.5 + -0.5 + 15x + -15x + -5x2 + 5x2 Combine like terms: 0.5 + -0.5 = 0.0 9.5 + -15x + 5x2 = 0.0 + 15x + -15x + -5x2 + 5x2 9.5 + -15x + 5x2 = 15x + -15x + -5x2 + 5x2 Combine like terms: 15x + -15x = 0 9.5 + -15x + 5x2 = 0 + -5x2 + 5x2 9.5 + -15x + 5x2 = -5x2 + 5x2 Combine like terms: -5x2 + 5x2 = 0 9.5 + -15x + 5x2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 1.9 + -3x + x2 = 0 Move the constant term to the right: Add '-1.9' to each side of the equation. 1.9 + -3x + -1.9 + x2 = 0 + -1.9 Reorder the terms: 1.9 + -1.9 + -3x + x2 = 0 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + -3x + x2 = 0 + -1.9 -3x + x2 = 0 + -1.9 Combine like terms: 0 + -1.9 = -1.9 -3x + x2 = -1.9 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 = -1.9 + 2.25 Reorder the terms: 2.25 + -3x + x2 = -1.9 + 2.25 Combine like terms: -1.9 + 2.25 = 0.35 2.25 + -3x + x2 = 0.35 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 0.35 Calculate the square root of the right side: 0.591607978 Break this problem into two subproblems by setting (x + -1.5) equal to 0.591607978 and -0.591607978.Subproblem 1
x + -1.5 = 0.591607978 Simplifying x + -1.5 = 0.591607978 Reorder the terms: -1.5 + x = 0.591607978 Solving -1.5 + x = 0.591607978 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 = 0.591607978 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 0.591607978 + 1.5 x = 0.591607978 + 1.5 Combine like terms: 0.591607978 + 1.5 = 2.091607978 x = 2.091607978 Simplifying x = 2.091607978Subproblem 2
x + -1.5 = -0.591607978 Simplifying x + -1.5 = -0.591607978 Reorder the terms: -1.5 + x = -0.591607978 Solving -1.5 + x = -0.591607978 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 = -0.591607978 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -0.591607978 + 1.5 x = -0.591607978 + 1.5 Combine like terms: -0.591607978 + 1.5 = 0.908392022 x = 0.908392022 Simplifying x = 0.908392022Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.091607978, 0.908392022}
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