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Simplifying 24x2 + -27x + 7 = 0 Reorder the terms: 7 + -27x + 24x2 = 0 Solving 7 + -27x + 24x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. 0.2916666667 + -1.125x + x2 = 0 Move the constant term to the right: Add '-0.2916666667' to each side of the equation. 0.2916666667 + -1.125x + -0.2916666667 + x2 = 0 + -0.2916666667 Reorder the terms: 0.2916666667 + -0.2916666667 + -1.125x + x2 = 0 + -0.2916666667 Combine like terms: 0.2916666667 + -0.2916666667 = 0.0000000000 0.0000000000 + -1.125x + x2 = 0 + -0.2916666667 -1.125x + x2 = 0 + -0.2916666667 Combine like terms: 0 + -0.2916666667 = -0.2916666667 -1.125x + x2 = -0.2916666667 The x term is -1.125x. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125x + 0.31640625 + x2 = -0.2916666667 + 0.31640625 Reorder the terms: 0.31640625 + -1.125x + x2 = -0.2916666667 + 0.31640625 Combine like terms: -0.2916666667 + 0.31640625 = 0.0247395833 0.31640625 + -1.125x + x2 = 0.0247395833 Factor a perfect square on the left side: (x + -0.5625)(x + -0.5625) = 0.0247395833 Calculate the square root of the right side: 0.157288217 Break this problem into two subproblems by setting (x + -0.5625) equal to 0.157288217 and -0.157288217.Subproblem 1
x + -0.5625 = 0.157288217 Simplifying x + -0.5625 = 0.157288217 Reorder the terms: -0.5625 + x = 0.157288217 Solving -0.5625 + x = 0.157288217 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + x = 0.157288217 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + x = 0.157288217 + 0.5625 x = 0.157288217 + 0.5625 Combine like terms: 0.157288217 + 0.5625 = 0.719788217 x = 0.719788217 Simplifying x = 0.719788217Subproblem 2
x + -0.5625 = -0.157288217 Simplifying x + -0.5625 = -0.157288217 Reorder the terms: -0.5625 + x = -0.157288217 Solving -0.5625 + x = -0.157288217 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + x = -0.157288217 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + x = -0.157288217 + 0.5625 x = -0.157288217 + 0.5625 Combine like terms: -0.157288217 + 0.5625 = 0.405211783 x = 0.405211783 Simplifying x = 0.405211783Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.719788217, 0.405211783}
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