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Simplifying 54x3 + 46x2 + -24x = 0 Reorder the terms: -24x + 46x2 + 54x3 = 0 Solving -24x + 46x2 + 54x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2x'. 2x(-12 + 23x + 27x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(-12 + 23x + 27x2)' equal to zero and attempt to solve: Simplifying -12 + 23x + 27x2 = 0 Solving -12 + 23x + 27x2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.4444444444 + 0.8518518519x + x2 = 0 Move the constant term to the right: Add '0.4444444444' to each side of the equation. -0.4444444444 + 0.8518518519x + 0.4444444444 + x2 = 0 + 0.4444444444 Reorder the terms: -0.4444444444 + 0.4444444444 + 0.8518518519x + x2 = 0 + 0.4444444444 Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000 0.0000000000 + 0.8518518519x + x2 = 0 + 0.4444444444 0.8518518519x + x2 = 0 + 0.4444444444 Combine like terms: 0 + 0.4444444444 = 0.4444444444 0.8518518519x + x2 = 0.4444444444 The x term is 0.8518518519x. Take half its coefficient (0.425925926). Square it (0.1814128944) and add it to both sides. Add '0.1814128944' to each side of the equation. 0.8518518519x + 0.1814128944 + x2 = 0.4444444444 + 0.1814128944 Reorder the terms: 0.1814128944 + 0.8518518519x + x2 = 0.4444444444 + 0.1814128944 Combine like terms: 0.4444444444 + 0.1814128944 = 0.6258573388 0.1814128944 + 0.8518518519x + x2 = 0.6258573388 Factor a perfect square on the left side: (x + 0.425925926)(x + 0.425925926) = 0.6258573388 Calculate the square root of the right side: 0.791111458 Break this problem into two subproblems by setting (x + 0.425925926) equal to 0.791111458 and -0.791111458.Subproblem 1
x + 0.425925926 = 0.791111458 Simplifying x + 0.425925926 = 0.791111458 Reorder the terms: 0.425925926 + x = 0.791111458 Solving 0.425925926 + x = 0.791111458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.425925926' to each side of the equation. 0.425925926 + -0.425925926 + x = 0.791111458 + -0.425925926 Combine like terms: 0.425925926 + -0.425925926 = 0.000000000 0.000000000 + x = 0.791111458 + -0.425925926 x = 0.791111458 + -0.425925926 Combine like terms: 0.791111458 + -0.425925926 = 0.365185532 x = 0.365185532 Simplifying x = 0.365185532Subproblem 2
x + 0.425925926 = -0.791111458 Simplifying x + 0.425925926 = -0.791111458 Reorder the terms: 0.425925926 + x = -0.791111458 Solving 0.425925926 + x = -0.791111458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.425925926' to each side of the equation. 0.425925926 + -0.425925926 + x = -0.791111458 + -0.425925926 Combine like terms: 0.425925926 + -0.425925926 = 0.000000000 0.000000000 + x = -0.791111458 + -0.425925926 x = -0.791111458 + -0.425925926 Combine like terms: -0.791111458 + -0.425925926 = -1.217037384 x = -1.217037384 Simplifying x = -1.217037384Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.365185532, -1.217037384}Solution
x = {0, 0.365185532, -1.217037384}
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