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Simplifying x2 + -0.5x + -10.5 = 0 Reorder the terms: -10.5 + -0.5x + x2 = 0 Solving -10.5 + -0.5x + x2 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + -0.5x + 10.5 + x2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + -0.5x + x2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + -0.5x + x2 = 0 + 10.5 -0.5x + x2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 -0.5x + x2 = 10.5 The x term is -0.5x. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5x + 0.0625 + x2 = 10.5 + 0.0625 Reorder the terms: 0.0625 + -0.5x + x2 = 10.5 + 0.0625 Combine like terms: 10.5 + 0.0625 = 10.5625 0.0625 + -0.5x + x2 = 10.5625 Factor a perfect square on the left side: (x + -0.25)(x + -0.25) = 10.5625 Calculate the square root of the right side: 3.25 Break this problem into two subproblems by setting (x + -0.25) equal to 3.25 and -3.25.Subproblem 1
x + -0.25 = 3.25 Simplifying x + -0.25 = 3.25 Reorder the terms: -0.25 + x = 3.25 Solving -0.25 + x = 3.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 3.25 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 3.25 + 0.25 x = 3.25 + 0.25 Combine like terms: 3.25 + 0.25 = 3.5 x = 3.5 Simplifying x = 3.5Subproblem 2
x + -0.25 = -3.25 Simplifying x + -0.25 = -3.25 Reorder the terms: -0.25 + x = -3.25 Solving -0.25 + x = -3.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -3.25 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -3.25 + 0.25 x = -3.25 + 0.25 Combine like terms: -3.25 + 0.25 = -3 x = -3 Simplifying x = -3Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.5, -3}
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