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Simplifying 27x2 + -78x + -8 = 0 Reorder the terms: -8 + -78x + 27x2 = 0 Solving -8 + -78x + 27x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.2962962963 + -2.888888889x + x2 = 0 Move the constant term to the right: Add '0.2962962963' to each side of the equation. -0.2962962963 + -2.888888889x + 0.2962962963 + x2 = 0 + 0.2962962963 Reorder the terms: -0.2962962963 + 0.2962962963 + -2.888888889x + x2 = 0 + 0.2962962963 Combine like terms: -0.2962962963 + 0.2962962963 = 0.0000000000 0.0000000000 + -2.888888889x + x2 = 0 + 0.2962962963 -2.888888889x + x2 = 0 + 0.2962962963 Combine like terms: 0 + 0.2962962963 = 0.2962962963 -2.888888889x + x2 = 0.2962962963 The x term is -2.888888889x. Take half its coefficient (-1.444444445). Square it (2.086419755) and add it to both sides. Add '2.086419755' to each side of the equation. -2.888888889x + 2.086419755 + x2 = 0.2962962963 + 2.086419755 Reorder the terms: 2.086419755 + -2.888888889x + x2 = 0.2962962963 + 2.086419755 Combine like terms: 0.2962962963 + 2.086419755 = 2.3827160513 2.086419755 + -2.888888889x + x2 = 2.3827160513 Factor a perfect square on the left side: (x + -1.444444445)(x + -1.444444445) = 2.3827160513 Calculate the square root of the right side: 1.543604888 Break this problem into two subproblems by setting (x + -1.444444445) equal to 1.543604888 and -1.543604888.Subproblem 1
x + -1.444444445 = 1.543604888 Simplifying x + -1.444444445 = 1.543604888 Reorder the terms: -1.444444445 + x = 1.543604888 Solving -1.444444445 + x = 1.543604888 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + x = 1.543604888 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + x = 1.543604888 + 1.444444445 x = 1.543604888 + 1.444444445 Combine like terms: 1.543604888 + 1.444444445 = 2.988049333 x = 2.988049333 Simplifying x = 2.988049333Subproblem 2
x + -1.444444445 = -1.543604888 Simplifying x + -1.444444445 = -1.543604888 Reorder the terms: -1.444444445 + x = -1.543604888 Solving -1.444444445 + x = -1.543604888 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + x = -1.543604888 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + x = -1.543604888 + 1.444444445 x = -1.543604888 + 1.444444445 Combine like terms: -1.543604888 + 1.444444445 = -0.099160443 x = -0.099160443 Simplifying x = -0.099160443Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.988049333, -0.099160443}
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