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Simplifying 3x2 + 5 = 9x2 + 6x + -1 Reorder the terms: 5 + 3x2 = 9x2 + 6x + -1 Reorder the terms: 5 + 3x2 = -1 + 6x + 9x2 Solving 5 + 3x2 = -1 + 6x + 9x2 Solving for variable 'x'. Reorder the terms: 5 + 1 + -6x + 3x2 + -9x2 = -1 + 6x + 9x2 + 1 + -6x + -9x2 Combine like terms: 5 + 1 = 6 6 + -6x + 3x2 + -9x2 = -1 + 6x + 9x2 + 1 + -6x + -9x2 Combine like terms: 3x2 + -9x2 = -6x2 6 + -6x + -6x2 = -1 + 6x + 9x2 + 1 + -6x + -9x2 Reorder the terms: 6 + -6x + -6x2 = -1 + 1 + 6x + -6x + 9x2 + -9x2 Combine like terms: -1 + 1 = 0 6 + -6x + -6x2 = 0 + 6x + -6x + 9x2 + -9x2 6 + -6x + -6x2 = 6x + -6x + 9x2 + -9x2 Combine like terms: 6x + -6x = 0 6 + -6x + -6x2 = 0 + 9x2 + -9x2 6 + -6x + -6x2 = 9x2 + -9x2 Combine like terms: 9x2 + -9x2 = 0 6 + -6x + -6x2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(1 + -1x + -1x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(1 + -1x + -1x2)' equal to zero and attempt to solve: Simplifying 1 + -1x + -1x2 = 0 Solving 1 + -1x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + x + x2 = 0 + 1 x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 x + x2 = 1 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = 1 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 1.25 + x2 = 1.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (x + 0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
x + 0.5 = 1.118033989 Simplifying x + 0.5 = 1.118033989 Reorder the terms: 0.5 + x = 1.118033989 Solving 0.5 + x = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.118033989 + -0.5 x = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 x = 0.618033989 Simplifying x = 0.618033989Subproblem 2
x + 0.5 = -1.118033989 Simplifying x + 0.5 = -1.118033989 Reorder the terms: 0.5 + x = -1.118033989 Solving 0.5 + x = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.118033989 + -0.5 x = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 x = -1.618033989 Simplifying x = -1.618033989Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.618033989, -1.618033989}Solution
x = {0.618033989, -1.618033989}
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