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Simplifying 0.25x2 + -8.25x + 54 = 0 Reorder the terms: 54 + -8.25x + 0.25x2 = 0 Solving 54 + -8.25x + 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'. 216 + -33x + x2 = 0 Move the constant term to the right: Add '-216' to each side of the equation. 216 + -33x + -216 + x2 = 0 + -216 Reorder the terms: 216 + -216 + -33x + x2 = 0 + -216 Combine like terms: 216 + -216 = 0 0 + -33x + x2 = 0 + -216 -33x + x2 = 0 + -216 Combine like terms: 0 + -216 = -216 -33x + x2 = -216 The x term is -33x. Take half its coefficient (-16.5). Square it (272.25) and add it to both sides. Add '272.25' to each side of the equation. -33x + 272.25 + x2 = -216 + 272.25 Reorder the terms: 272.25 + -33x + x2 = -216 + 272.25 Combine like terms: -216 + 272.25 = 56.25 272.25 + -33x + x2 = 56.25 Factor a perfect square on the left side: (x + -16.5)(x + -16.5) = 56.25 Calculate the square root of the right side: 7.5 Break this problem into two subproblems by setting (x + -16.5) equal to 7.5 and -7.5.Subproblem 1
x + -16.5 = 7.5 Simplifying x + -16.5 = 7.5 Reorder the terms: -16.5 + x = 7.5 Solving -16.5 + x = 7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16.5' to each side of the equation. -16.5 + 16.5 + x = 7.5 + 16.5 Combine like terms: -16.5 + 16.5 = 0.0 0.0 + x = 7.5 + 16.5 x = 7.5 + 16.5 Combine like terms: 7.5 + 16.5 = 24 x = 24 Simplifying x = 24Subproblem 2
x + -16.5 = -7.5 Simplifying x + -16.5 = -7.5 Reorder the terms: -16.5 + x = -7.5 Solving -16.5 + x = -7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '16.5' to each side of the equation. -16.5 + 16.5 + x = -7.5 + 16.5 Combine like terms: -16.5 + 16.5 = 0.0 0.0 + x = -7.5 + 16.5 x = -7.5 + 16.5 Combine like terms: -7.5 + 16.5 = 9 x = 9 Simplifying x = 9Solution
The solution to the problem is based on the solutions from the subproblems. x = {24, 9}
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