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Simplifying 180 = x2 + -6x + 0.5x + 42 Reorder the terms: 180 = 42 + -6x + 0.5x + x2 Combine like terms: -6x + 0.5x = -5.5x 180 = 42 + -5.5x + x2 Solving 180 = 42 + -5.5x + x2 Solving for variable 'x'. Combine like terms: 180 + -42 = 138 138 + 5.5x + -1x2 = 42 + -5.5x + x2 + -42 + 5.5x + -1x2 Reorder the terms: 138 + 5.5x + -1x2 = 42 + -42 + -5.5x + 5.5x + x2 + -1x2 Combine like terms: 42 + -42 = 0 138 + 5.5x + -1x2 = 0 + -5.5x + 5.5x + x2 + -1x2 138 + 5.5x + -1x2 = -5.5x + 5.5x + x2 + -1x2 Combine like terms: -5.5x + 5.5x = 0.0 138 + 5.5x + -1x2 = 0.0 + x2 + -1x2 138 + 5.5x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 138 + 5.5x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -138 + -5.5x + x2 = 0 Move the constant term to the right: Add '138' to each side of the equation. -138 + -5.5x + 138 + x2 = 0 + 138 Reorder the terms: -138 + 138 + -5.5x + x2 = 0 + 138 Combine like terms: -138 + 138 = 0 0 + -5.5x + x2 = 0 + 138 -5.5x + x2 = 0 + 138 Combine like terms: 0 + 138 = 138 -5.5x + x2 = 138 The x term is -5.5x. Take half its coefficient (-2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. -5.5x + 7.5625 + x2 = 138 + 7.5625 Reorder the terms: 7.5625 + -5.5x + x2 = 138 + 7.5625 Combine like terms: 138 + 7.5625 = 145.5625 7.5625 + -5.5x + x2 = 145.5625 Factor a perfect square on the left side: (x + -2.75)(x + -2.75) = 145.5625 Calculate the square root of the right side: 12.064928512 Break this problem into two subproblems by setting (x + -2.75) equal to 12.064928512 and -12.064928512.Subproblem 1
x + -2.75 = 12.064928512 Simplifying x + -2.75 = 12.064928512 Reorder the terms: -2.75 + x = 12.064928512 Solving -2.75 + x = 12.064928512 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + x = 12.064928512 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + x = 12.064928512 + 2.75 x = 12.064928512 + 2.75 Combine like terms: 12.064928512 + 2.75 = 14.814928512 x = 14.814928512 Simplifying x = 14.814928512Subproblem 2
x + -2.75 = -12.064928512 Simplifying x + -2.75 = -12.064928512 Reorder the terms: -2.75 + x = -12.064928512 Solving -2.75 + x = -12.064928512 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + x = -12.064928512 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + x = -12.064928512 + 2.75 x = -12.064928512 + 2.75 Combine like terms: -12.064928512 + 2.75 = -9.314928512 x = -9.314928512 Simplifying x = -9.314928512Solution
The solution to the problem is based on the solutions from the subproblems. x = {14.814928512, -9.314928512}
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