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Simplifying 0.25x2 + 6.25 + 2.5x = 25 + -1x2 Reorder the terms: 6.25 + 2.5x + 0.25x2 = 25 + -1x2 Solving 6.25 + 2.5x + 0.25x2 = 25 + -1x2 Solving for variable 'x'. Reorder the terms: 6.25 + -25 + 2.5x + 0.25x2 + x2 = 25 + -1x2 + -25 + x2 Combine like terms: 6.25 + -25 = -18.75 -18.75 + 2.5x + 0.25x2 + x2 = 25 + -1x2 + -25 + x2 Combine like terms: 0.25x2 + x2 = 1.25x2 -18.75 + 2.5x + 1.25x2 = 25 + -1x2 + -25 + x2 Reorder the terms: -18.75 + 2.5x + 1.25x2 = 25 + -25 + -1x2 + x2 Combine like terms: 25 + -25 = 0 -18.75 + 2.5x + 1.25x2 = 0 + -1x2 + x2 -18.75 + 2.5x + 1.25x2 = -1x2 + x2 Combine like terms: -1x2 + x2 = 0 -18.75 + 2.5x + 1.25x2 = 0 Begin completing the square. Divide all terms by 1.25 the coefficient of the squared term: Divide each side by '1.25'. -15 + 2x + x2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + 2x + 15 + x2 = 0 + 15 Reorder the terms: -15 + 15 + 2x + x2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 2x + x2 = 0 + 15 2x + x2 = 0 + 15 Combine like terms: 0 + 15 = 15 2x + x2 = 15 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 15 + 1 Reorder the terms: 1 + 2x + x2 = 15 + 1 Combine like terms: 15 + 1 = 16 1 + 2x + x2 = 16 Factor a perfect square on the left side: (x + 1)(x + 1) = 16 Calculate the square root of the right side: 4 Break this problem into two subproblems by setting (x + 1) equal to 4 and -4.Subproblem 1
x + 1 = 4 Simplifying x + 1 = 4 Reorder the terms: 1 + x = 4 Solving 1 + x = 4 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 4 + -1 Combine like terms: 1 + -1 = 0 0 + x = 4 + -1 x = 4 + -1 Combine like terms: 4 + -1 = 3 x = 3 Simplifying x = 3Subproblem 2
x + 1 = -4 Simplifying x + 1 = -4 Reorder the terms: 1 + x = -4 Solving 1 + x = -4 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -4 + -1 Combine like terms: 1 + -1 = 0 0 + x = -4 + -1 x = -4 + -1 Combine like terms: -4 + -1 = -5 x = -5 Simplifying x = -5Solution
The solution to the problem is based on the solutions from the subproblems. x = {3, -5}
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