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Simplifying 6x2 + 19x + -69 = 0 Reorder the terms: -69 + 19x + 6x2 = 0 Solving -69 + 19x + 6x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -11.5 + 3.166666667x + x2 = 0 Move the constant term to the right: Add '11.5' to each side of the equation. -11.5 + 3.166666667x + 11.5 + x2 = 0 + 11.5 Reorder the terms: -11.5 + 11.5 + 3.166666667x + x2 = 0 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + 3.166666667x + x2 = 0 + 11.5 3.166666667x + x2 = 0 + 11.5 Combine like terms: 0 + 11.5 = 11.5 3.166666667x + x2 = 11.5 The x term is 3.166666667x. Take half its coefficient (1.583333334). Square it (2.506944447) and add it to both sides. Add '2.506944447' to each side of the equation. 3.166666667x + 2.506944447 + x2 = 11.5 + 2.506944447 Reorder the terms: 2.506944447 + 3.166666667x + x2 = 11.5 + 2.506944447 Combine like terms: 11.5 + 2.506944447 = 14.006944447 2.506944447 + 3.166666667x + x2 = 14.006944447 Factor a perfect square on the left side: (x + 1.583333334)(x + 1.583333334) = 14.006944447 Calculate the square root of the right side: 3.742585262 Break this problem into two subproblems by setting (x + 1.583333334) equal to 3.742585262 and -3.742585262.Subproblem 1
x + 1.583333334 = 3.742585262 Simplifying x + 1.583333334 = 3.742585262 Reorder the terms: 1.583333334 + x = 3.742585262 Solving 1.583333334 + x = 3.742585262 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + x = 3.742585262 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = 3.742585262 + -1.583333334 x = 3.742585262 + -1.583333334 Combine like terms: 3.742585262 + -1.583333334 = 2.159251928 x = 2.159251928 Simplifying x = 2.159251928Subproblem 2
x + 1.583333334 = -3.742585262 Simplifying x + 1.583333334 = -3.742585262 Reorder the terms: 1.583333334 + x = -3.742585262 Solving 1.583333334 + x = -3.742585262 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + x = -3.742585262 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + x = -3.742585262 + -1.583333334 x = -3.742585262 + -1.583333334 Combine like terms: -3.742585262 + -1.583333334 = -5.325918596 x = -5.325918596 Simplifying x = -5.325918596Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.159251928, -5.325918596}
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