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Simplifying 0 = 5x2 + 10x + 3 Reorder the terms: 0 = 3 + 10x + 5x2 Solving 0 = 3 + 10x + 5x2 Solving for variable 'x'. Combine like terms: 0 + -3 = -3 -3 + -10x + -5x2 = 3 + 10x + 5x2 + -3 + -10x + -5x2 Reorder the terms: -3 + -10x + -5x2 = 3 + -3 + 10x + -10x + 5x2 + -5x2 Combine like terms: 3 + -3 = 0 -3 + -10x + -5x2 = 0 + 10x + -10x + 5x2 + -5x2 -3 + -10x + -5x2 = 10x + -10x + 5x2 + -5x2 Combine like terms: 10x + -10x = 0 -3 + -10x + -5x2 = 0 + 5x2 + -5x2 -3 + -10x + -5x2 = 5x2 + -5x2 Combine like terms: 5x2 + -5x2 = 0 -3 + -10x + -5x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(3 + 10x + 5x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(3 + 10x + 5x2)' equal to zero and attempt to solve: Simplifying 3 + 10x + 5x2 = 0 Solving 3 + 10x + 5x2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.6 + 2x + x2 = 0 Move the constant term to the right: Add '-0.6' to each side of the equation. 0.6 + 2x + -0.6 + x2 = 0 + -0.6 Reorder the terms: 0.6 + -0.6 + 2x + x2 = 0 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + 2x + x2 = 0 + -0.6 2x + x2 = 0 + -0.6 Combine like terms: 0 + -0.6 = -0.6 2x + x2 = -0.6 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = -0.6 + 1 Reorder the terms: 1 + 2x + x2 = -0.6 + 1 Combine like terms: -0.6 + 1 = 0.4 1 + 2x + x2 = 0.4 Factor a perfect square on the left side: (x + 1)(x + 1) = 0.4 Calculate the square root of the right side: 0.632455532 Break this problem into two subproblems by setting (x + 1) equal to 0.632455532 and -0.632455532.Subproblem 1
x + 1 = 0.632455532 Simplifying x + 1 = 0.632455532 Reorder the terms: 1 + x = 0.632455532 Solving 1 + x = 0.632455532 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 0.632455532 + -1 Combine like terms: 1 + -1 = 0 0 + x = 0.632455532 + -1 x = 0.632455532 + -1 Combine like terms: 0.632455532 + -1 = -0.367544468 x = -0.367544468 Simplifying x = -0.367544468Subproblem 2
x + 1 = -0.632455532 Simplifying x + 1 = -0.632455532 Reorder the terms: 1 + x = -0.632455532 Solving 1 + x = -0.632455532 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -0.632455532 + -1 Combine like terms: 1 + -1 = 0 0 + x = -0.632455532 + -1 x = -0.632455532 + -1 Combine like terms: -0.632455532 + -1 = -1.632455532 x = -1.632455532 Simplifying x = -1.632455532Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.367544468, -1.632455532}Solution
x = {-0.367544468, -1.632455532}
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