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Simplifying 247 = 0.5(x + 7) * x Reorder the terms: 247 = 0.5(7 + x) * x Reorder the terms for easier multiplication: 247 = 0.5x(7 + x) 247 = (7 * 0.5x + x * 0.5x) 247 = (3.5x + 0.5x2) Solving 247 = 3.5x + 0.5x2 Solving for variable 'x'. Reorder the terms: 247 + -3.5x + -0.5x2 = 3.5x + -3.5x + 0.5x2 + -0.5x2 Combine like terms: 3.5x + -3.5x = 0.0 247 + -3.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 247 + -3.5x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 247 + -3.5x + -0.5x2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -494 + 7x + x2 = 0 Move the constant term to the right: Add '494' to each side of the equation. -494 + 7x + 494 + x2 = 0 + 494 Reorder the terms: -494 + 494 + 7x + x2 = 0 + 494 Combine like terms: -494 + 494 = 0 0 + 7x + x2 = 0 + 494 7x + x2 = 0 + 494 Combine like terms: 0 + 494 = 494 7x + x2 = 494 The x term is 7x. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7x + 12.25 + x2 = 494 + 12.25 Reorder the terms: 12.25 + 7x + x2 = 494 + 12.25 Combine like terms: 494 + 12.25 = 506.25 12.25 + 7x + x2 = 506.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 506.25 Calculate the square root of the right side: 22.5 Break this problem into two subproblems by setting (x + 3.5) equal to 22.5 and -22.5.Subproblem 1
x + 3.5 = 22.5 Simplifying x + 3.5 = 22.5 Reorder the terms: 3.5 + x = 22.5 Solving 3.5 + x = 22.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 22.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 22.5 + -3.5 x = 22.5 + -3.5 Combine like terms: 22.5 + -3.5 = 19 x = 19 Simplifying x = 19Subproblem 2
x + 3.5 = -22.5 Simplifying x + 3.5 = -22.5 Reorder the terms: 3.5 + x = -22.5 Solving 3.5 + x = -22.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -22.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -22.5 + -3.5 x = -22.5 + -3.5 Combine like terms: -22.5 + -3.5 = -26 x = -26 Simplifying x = -26Solution
The solution to the problem is based on the solutions from the subproblems. x = {19, -26}
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