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Simplifying -6x2 + 9x + 3 = 0 Reorder the terms: 3 + 9x + -6x2 = 0 Solving 3 + 9x + -6x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3'. 3(1 + 3x + -2x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(1 + 3x + -2x2)' equal to zero and attempt to solve: Simplifying 1 + 3x + -2x2 = 0 Solving 1 + 3x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + -1.5x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1.5x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1.5x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1.5x + x2 = 0 + 0.5 -1.5x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1.5x + x2 = 0.5 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = 0.5 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = 0.5 + 0.5625 Combine like terms: 0.5 + 0.5625 = 1.0625 0.5625 + -1.5x + x2 = 1.0625 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (x + -0.75) equal to 1.030776406 and -1.030776406.Subproblem 1
x + -0.75 = 1.030776406 Simplifying x + -0.75 = 1.030776406 Reorder the terms: -0.75 + x = 1.030776406 Solving -0.75 + x = 1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.030776406 + 0.75 x = 1.030776406 + 0.75 Combine like terms: 1.030776406 + 0.75 = 1.780776406 x = 1.780776406 Simplifying x = 1.780776406Subproblem 2
x + -0.75 = -1.030776406 Simplifying x + -0.75 = -1.030776406 Reorder the terms: -0.75 + x = -1.030776406 Solving -0.75 + x = -1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.030776406 + 0.75 x = -1.030776406 + 0.75 Combine like terms: -1.030776406 + 0.75 = -0.280776406 x = -0.280776406 Simplifying x = -0.280776406Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.780776406, -0.280776406}Solution
x = {1.780776406, -0.280776406}
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