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Simplifying 120 + -6x = 0.5x2 + x Reorder the terms: 120 + -6x = x + 0.5x2 Solving 120 + -6x = x + 0.5x2 Solving for variable 'x'. Combine like terms: -6x + -1x = -7x 120 + -7x + -0.5x2 = x + 0.5x2 + -1x + -0.5x2 Reorder the terms: 120 + -7x + -0.5x2 = x + -1x + 0.5x2 + -0.5x2 Combine like terms: x + -1x = 0 120 + -7x + -0.5x2 = 0 + 0.5x2 + -0.5x2 120 + -7x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 120 + -7x + -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'. -240 + 14x + x2 = 0 Move the constant term to the right: Add '240' to each side of the equation. -240 + 14x + 240 + x2 = 0 + 240 Reorder the terms: -240 + 240 + 14x + x2 = 0 + 240 Combine like terms: -240 + 240 = 0 0 + 14x + x2 = 0 + 240 14x + x2 = 0 + 240 Combine like terms: 0 + 240 = 240 14x + x2 = 240 The x term is 14x. Take half its coefficient (7). Square it (49) and add it to both sides. Add '49' to each side of the equation. 14x + 49 + x2 = 240 + 49 Reorder the terms: 49 + 14x + x2 = 240 + 49 Combine like terms: 240 + 49 = 289 49 + 14x + x2 = 289 Factor a perfect square on the left side: (x + 7)(x + 7) = 289 Calculate the square root of the right side: 17 Break this problem into two subproblems by setting (x + 7) equal to 17 and -17.Subproblem 1
x + 7 = 17 Simplifying x + 7 = 17 Reorder the terms: 7 + x = 17 Solving 7 + x = 17 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + x = 17 + -7 Combine like terms: 7 + -7 = 0 0 + x = 17 + -7 x = 17 + -7 Combine like terms: 17 + -7 = 10 x = 10 Simplifying x = 10Subproblem 2
x + 7 = -17 Simplifying x + 7 = -17 Reorder the terms: 7 + x = -17 Solving 7 + x = -17 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + x = -17 + -7 Combine like terms: 7 + -7 = 0 0 + x = -17 + -7 x = -17 + -7 Combine like terms: -17 + -7 = -24 x = -24 Simplifying x = -24Solution
The solution to the problem is based on the solutions from the subproblems. x = {10, -24}
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