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Simplifying x(x + 2) = 5x(x + 2) + -10 Reorder the terms: x(2 + x) = 5x(x + 2) + -10 (2 * x + x * x) = 5x(x + 2) + -10 (2x + x2) = 5x(x + 2) + -10 Reorder the terms: 2x + x2 = 5x(2 + x) + -10 2x + x2 = (2 * 5x + x * 5x) + -10 2x + x2 = (10x + 5x2) + -10 Reorder the terms: 2x + x2 = -10 + 10x + 5x2 Solving 2x + x2 = -10 + 10x + 5x2 Solving for variable 'x'. Reorder the terms: 10 + 2x + -10x + x2 + -5x2 = -10 + 10x + 5x2 + 10 + -10x + -5x2 Combine like terms: 2x + -10x = -8x 10 + -8x + x2 + -5x2 = -10 + 10x + 5x2 + 10 + -10x + -5x2 Combine like terms: x2 + -5x2 = -4x2 10 + -8x + -4x2 = -10 + 10x + 5x2 + 10 + -10x + -5x2 Reorder the terms: 10 + -8x + -4x2 = -10 + 10 + 10x + -10x + 5x2 + -5x2 Combine like terms: -10 + 10 = 0 10 + -8x + -4x2 = 0 + 10x + -10x + 5x2 + -5x2 10 + -8x + -4x2 = 10x + -10x + 5x2 + -5x2 Combine like terms: 10x + -10x = 0 10 + -8x + -4x2 = 0 + 5x2 + -5x2 10 + -8x + -4x2 = 5x2 + -5x2 Combine like terms: 5x2 + -5x2 = 0 10 + -8x + -4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(5 + -4x + -2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(5 + -4x + -2x2)' equal to zero and attempt to solve: Simplifying 5 + -4x + -2x2 = 0 Solving 5 + -4x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + 2x + x2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 2x + 2.5 + x2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 2x + x2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 2x + x2 = 0 + 2.5 2x + x2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 2x + x2 = 2.5 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 = 2.5 + 1 Reorder the terms: 1 + 2x + x2 = 2.5 + 1 Combine like terms: 2.5 + 1 = 3.5 1 + 2x + x2 = 3.5 Factor a perfect square on the left side: (x + 1)(x + 1) = 3.5 Calculate the square root of the right side: 1.870828693 Break this problem into two subproblems by setting (x + 1) equal to 1.870828693 and -1.870828693.Subproblem 1
x + 1 = 1.870828693 Simplifying x + 1 = 1.870828693 Reorder the terms: 1 + x = 1.870828693 Solving 1 + x = 1.870828693 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 = 1.870828693 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.870828693 + -1 x = 1.870828693 + -1 Combine like terms: 1.870828693 + -1 = 0.870828693 x = 0.870828693 Simplifying x = 0.870828693Subproblem 2
x + 1 = -1.870828693 Simplifying x + 1 = -1.870828693 Reorder the terms: 1 + x = -1.870828693 Solving 1 + x = -1.870828693 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 = -1.870828693 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.870828693 + -1 x = -1.870828693 + -1 Combine like terms: -1.870828693 + -1 = -2.870828693 x = -2.870828693 Simplifying x = -2.870828693Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.870828693, -2.870828693}Solution
x = {0.870828693, -2.870828693}
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