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Simplifying 2x2 + 16x + 24 = 212.5 Reorder the terms: 24 + 16x + 2x2 = 212.5 Solving 24 + 16x + 2x2 = 212.5 Solving for variable 'x'. Reorder the terms: 24 + -212.5 + 16x + 2x2 = 212.5 + -212.5 Combine like terms: 24 + -212.5 = -188.5 -188.5 + 16x + 2x2 = 212.5 + -212.5 Combine like terms: 212.5 + -212.5 = 0.0 -188.5 + 16x + 2x2 = 0.0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -94.25 + 8x + x2 = 0.0 Move the constant term to the right: Add '94.25' to each side of the equation. -94.25 + 8x + 94.25 + x2 = 0.0 + 94.25 Reorder the terms: -94.25 + 94.25 + 8x + x2 = 0.0 + 94.25 Combine like terms: -94.25 + 94.25 = 0.00 0.00 + 8x + x2 = 0.0 + 94.25 8x + x2 = 0.0 + 94.25 Combine like terms: 0.0 + 94.25 = 94.25 8x + x2 = 94.25 The x term is 8x. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8x + 16 + x2 = 94.25 + 16 Reorder the terms: 16 + 8x + x2 = 94.25 + 16 Combine like terms: 94.25 + 16 = 110.25 16 + 8x + x2 = 110.25 Factor a perfect square on the left side: (x + 4)(x + 4) = 110.25 Calculate the square root of the right side: 10.5 Break this problem into two subproblems by setting (x + 4) equal to 10.5 and -10.5.Subproblem 1
x + 4 = 10.5 Simplifying x + 4 = 10.5 Reorder the terms: 4 + x = 10.5 Solving 4 + x = 10.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = 10.5 + -4 Combine like terms: 4 + -4 = 0 0 + x = 10.5 + -4 x = 10.5 + -4 Combine like terms: 10.5 + -4 = 6.5 x = 6.5 Simplifying x = 6.5Subproblem 2
x + 4 = -10.5 Simplifying x + 4 = -10.5 Reorder the terms: 4 + x = -10.5 Solving 4 + x = -10.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = -10.5 + -4 Combine like terms: 4 + -4 = 0 0 + x = -10.5 + -4 x = -10.5 + -4 Combine like terms: -10.5 + -4 = -14.5 x = -14.5 Simplifying x = -14.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.5, -14.5}
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