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Simplifying 12x2 + -21x + 6.5 = 0 Reorder the terms: 6.5 + -21x + 12x2 = 0 Solving 6.5 + -21x + 12x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.5416666667 + -1.75x + x2 = 0 Move the constant term to the right: Add '-0.5416666667' to each side of the equation. 0.5416666667 + -1.75x + -0.5416666667 + x2 = 0 + -0.5416666667 Reorder the terms: 0.5416666667 + -0.5416666667 + -1.75x + x2 = 0 + -0.5416666667 Combine like terms: 0.5416666667 + -0.5416666667 = 0.0000000000 0.0000000000 + -1.75x + x2 = 0 + -0.5416666667 -1.75x + x2 = 0 + -0.5416666667 Combine like terms: 0 + -0.5416666667 = -0.5416666667 -1.75x + x2 = -0.5416666667 The x term is -1.75x. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75x + 0.765625 + x2 = -0.5416666667 + 0.765625 Reorder the terms: 0.765625 + -1.75x + x2 = -0.5416666667 + 0.765625 Combine like terms: -0.5416666667 + 0.765625 = 0.2239583333 0.765625 + -1.75x + x2 = 0.2239583333 Factor a perfect square on the left side: (x + -0.875)(x + -0.875) = 0.2239583333 Calculate the square root of the right side: 0.473242362 Break this problem into two subproblems by setting (x + -0.875) equal to 0.473242362 and -0.473242362.Subproblem 1
x + -0.875 = 0.473242362 Simplifying x + -0.875 = 0.473242362 Reorder the terms: -0.875 + x = 0.473242362 Solving -0.875 + x = 0.473242362 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = 0.473242362 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = 0.473242362 + 0.875 x = 0.473242362 + 0.875 Combine like terms: 0.473242362 + 0.875 = 1.348242362 x = 1.348242362 Simplifying x = 1.348242362Subproblem 2
x + -0.875 = -0.473242362 Simplifying x + -0.875 = -0.473242362 Reorder the terms: -0.875 + x = -0.473242362 Solving -0.875 + x = -0.473242362 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = -0.473242362 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = -0.473242362 + 0.875 x = -0.473242362 + 0.875 Combine like terms: -0.473242362 + 0.875 = 0.401757638 x = 0.401757638 Simplifying x = 0.401757638Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.348242362, 0.401757638}
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