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Simplifying 0.25x2 + -0.25x + -10.5 = 0 Reorder the terms: -10.5 + -0.25x + 0.25x2 = 0 Solving -10.5 + -0.25x + 0.25x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.25 the coefficient of the squared term: Divide each side by '0.25'. -42 + -1x + x2 = 0 Move the constant term to the right: Add '42' to each side of the equation. -42 + -1x + 42 + x2 = 0 + 42 Reorder the terms: -42 + 42 + -1x + x2 = 0 + 42 Combine like terms: -42 + 42 = 0 0 + -1x + x2 = 0 + 42 -1x + x2 = 0 + 42 Combine like terms: 0 + 42 = 42 -1x + x2 = 42 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = 42 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 42 + 0.25 Combine like terms: 42 + 0.25 = 42.25 0.25 + -1x + x2 = 42.25 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 42.25 Calculate the square root of the right side: 6.5 Break this problem into two subproblems by setting (x + -0.5) equal to 6.5 and -6.5.Subproblem 1
x + -0.5 = 6.5 Simplifying x + -0.5 = 6.5 Reorder the terms: -0.5 + x = 6.5 Solving -0.5 + x = 6.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 6.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 6.5 + 0.5 x = 6.5 + 0.5 Combine like terms: 6.5 + 0.5 = 7 x = 7 Simplifying x = 7Subproblem 2
x + -0.5 = -6.5 Simplifying x + -0.5 = -6.5 Reorder the terms: -0.5 + x = -6.5 Solving -0.5 + x = -6.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -6.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -6.5 + 0.5 x = -6.5 + 0.5 Combine like terms: -6.5 + 0.5 = -6 x = -6 Simplifying x = -6Solution
The solution to the problem is based on the solutions from the subproblems. x = {7, -6}
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