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Simplifying 0 = 7.7x2 + -7.7x + 1 Reorder the terms: 0 = 1 + -7.7x + 7.7x2 Solving 0 = 1 + -7.7x + 7.7x2 Solving for variable 'x'. Combine like terms: 0 + -1 = -1 -1 + 7.7x + -7.7x2 = 1 + -7.7x + 7.7x2 + -1 + 7.7x + -7.7x2 Reorder the terms: -1 + 7.7x + -7.7x2 = 1 + -1 + -7.7x + 7.7x + 7.7x2 + -7.7x2 Combine like terms: 1 + -1 = 0 -1 + 7.7x + -7.7x2 = 0 + -7.7x + 7.7x + 7.7x2 + -7.7x2 -1 + 7.7x + -7.7x2 = -7.7x + 7.7x + 7.7x2 + -7.7x2 Combine like terms: -7.7x + 7.7x = 0.0 -1 + 7.7x + -7.7x2 = 0.0 + 7.7x2 + -7.7x2 -1 + 7.7x + -7.7x2 = 7.7x2 + -7.7x2 Combine like terms: 7.7x2 + -7.7x2 = 0.0 -1 + 7.7x + -7.7x2 = 0.0 Begin completing the square. Divide all terms by -7.7 the coefficient of the squared term: Divide each side by '-7.7'. 0.1298701299 + -1x + x2 = 0 Move the constant term to the right: Add '-0.1298701299' to each side of the equation. 0.1298701299 + -1x + -0.1298701299 + x2 = 0 + -0.1298701299 Reorder the terms: 0.1298701299 + -0.1298701299 + -1x + x2 = 0 + -0.1298701299 Combine like terms: 0.1298701299 + -0.1298701299 = 0.0000000000 0.0000000000 + -1x + x2 = 0 + -0.1298701299 -1x + x2 = 0 + -0.1298701299 Combine like terms: 0 + -0.1298701299 = -0.1298701299 -1x + x2 = -0.1298701299 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = -0.1298701299 + 0.25 Reorder the terms: 0.25 + -1x + x2 = -0.1298701299 + 0.25 Combine like terms: -0.1298701299 + 0.25 = 0.1201298701 0.25 + -1x + x2 = 0.1201298701 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.1201298701 Calculate the square root of the right side: 0.346597562 Break this problem into two subproblems by setting (x + -0.5) equal to 0.346597562 and -0.346597562.Subproblem 1
x + -0.5 = 0.346597562 Simplifying x + -0.5 = 0.346597562 Reorder the terms: -0.5 + x = 0.346597562 Solving -0.5 + x = 0.346597562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 0.346597562 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.346597562 + 0.5 x = 0.346597562 + 0.5 Combine like terms: 0.346597562 + 0.5 = 0.846597562 x = 0.846597562 Simplifying x = 0.846597562Subproblem 2
x + -0.5 = -0.346597562 Simplifying x + -0.5 = -0.346597562 Reorder the terms: -0.5 + x = -0.346597562 Solving -0.5 + x = -0.346597562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -0.346597562 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.346597562 + 0.5 x = -0.346597562 + 0.5 Combine like terms: -0.346597562 + 0.5 = 0.153402438 x = 0.153402438 Simplifying x = 0.153402438Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.846597562, 0.153402438}
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