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Simplifying 5x + 10 = (3 + 2x) * 5x Reorder the terms: 10 + 5x = (3 + 2x) * 5x Reorder the terms for easier multiplication: 10 + 5x = 5x(3 + 2x) 10 + 5x = (3 * 5x + 2x * 5x) 10 + 5x = (15x + 10x2) Solving 10 + 5x = 15x + 10x2 Solving for variable 'x'. Combine like terms: 5x + -15x = -10x 10 + -10x + -10x2 = 15x + 10x2 + -15x + -10x2 Reorder the terms: 10 + -10x + -10x2 = 15x + -15x + 10x2 + -10x2 Combine like terms: 15x + -15x = 0 10 + -10x + -10x2 = 0 + 10x2 + -10x2 10 + -10x + -10x2 = 10x2 + -10x2 Combine like terms: 10x2 + -10x2 = 0 10 + -10x + -10x2 = 0 Factor out the Greatest Common Factor (GCF), '10'. 10(1 + -1x + -1x2) = 0 Ignore the factor 10.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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