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Simplifying 0.5x(x + 7) = 49 Reorder the terms: 0.5x(7 + x) = 49 (7 * 0.5x + x * 0.5x) = 49 (3.5x + 0.5x2) = 49 Solving 3.5x + 0.5x2 = 49 Solving for variable 'x'. Reorder the terms: -49 + 3.5x + 0.5x2 = 49 + -49 Combine like terms: 49 + -49 = 0 -49 + 3.5x + 0.5x2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -98 + 7x + x2 = 0 Move the constant term to the right: Add '98' to each side of the equation. -98 + 7x + 98 + x2 = 0 + 98 Reorder the terms: -98 + 98 + 7x + x2 = 0 + 98 Combine like terms: -98 + 98 = 0 0 + 7x + x2 = 0 + 98 7x + x2 = 0 + 98 Combine like terms: 0 + 98 = 98 7x + x2 = 98 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 = 98 + 12.25 Reorder the terms: 12.25 + 7x + x2 = 98 + 12.25 Combine like terms: 98 + 12.25 = 110.25 12.25 + 7x + x2 = 110.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 110.25 Calculate the square root of the right side: 10.5 Break this problem into two subproblems by setting (x + 3.5) equal to 10.5 and -10.5.Subproblem 1
x + 3.5 = 10.5 Simplifying x + 3.5 = 10.5 Reorder the terms: 3.5 + x = 10.5 Solving 3.5 + x = 10.5 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 = 10.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 10.5 + -3.5 x = 10.5 + -3.5 Combine like terms: 10.5 + -3.5 = 7 x = 7 Simplifying x = 7Subproblem 2
x + 3.5 = -10.5 Simplifying x + 3.5 = -10.5 Reorder the terms: 3.5 + x = -10.5 Solving 3.5 + x = -10.5 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 = -10.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -10.5 + -3.5 x = -10.5 + -3.5 Combine like terms: -10.5 + -3.5 = -14 x = -14 Simplifying x = -14Solution
The solution to the problem is based on the solutions from the subproblems. x = {7, -14}
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