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Simplifying (x2 + x)(x + 6) = 7x Reorder the terms: (x + x2)(x + 6) = 7x Reorder the terms: (x + x2)(6 + x) = 7x Multiply (x + x2) * (6 + x) (x(6 + x) + x2(6 + x)) = 7x ((6 * x + x * x) + x2(6 + x)) = 7x ((6x + x2) + x2(6 + x)) = 7x (6x + x2 + (6 * x2 + x * x2)) = 7x (6x + x2 + (6x2 + x3)) = 7x Combine like terms: x2 + 6x2 = 7x2 (6x + 7x2 + x3) = 7x Solving 6x + 7x2 + x3 = 7x Solving for variable 'x'. Reorder the terms: 6x + -7x + 7x2 + x3 = 7x + -7x Combine like terms: 6x + -7x = -1x -1x + 7x2 + x3 = 7x + -7x Combine like terms: 7x + -7x = 0 -1x + 7x2 + x3 = 0 Factor out the Greatest Common Factor (GCF), 'x'. x(-1 + 7x + x2) = 0Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(-1 + 7x + x2)' equal to zero and attempt to solve: Simplifying -1 + 7x + x2 = 0 Solving -1 + 7x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + 7x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + 7x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 7x + x2 = 0 + 1 7x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 7x + x2 = 1 The x term is 7x. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7x + 12.25 + x2 = 1 + 12.25 Reorder the terms: 12.25 + 7x + x2 = 1 + 12.25 Combine like terms: 1 + 12.25 = 13.25 12.25 + 7x + x2 = 13.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 13.25 Calculate the square root of the right side: 3.640054945 Break this problem into two subproblems by setting (x + 3.5) equal to 3.640054945 and -3.640054945.Subproblem 1
x + 3.5 = 3.640054945 Simplifying x + 3.5 = 3.640054945 Reorder the terms: 3.5 + x = 3.640054945 Solving 3.5 + x = 3.640054945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 3.640054945 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 3.640054945 + -3.5 x = 3.640054945 + -3.5 Combine like terms: 3.640054945 + -3.5 = 0.140054945 x = 0.140054945 Simplifying x = 0.140054945Subproblem 2
x + 3.5 = -3.640054945 Simplifying x + 3.5 = -3.640054945 Reorder the terms: 3.5 + x = -3.640054945 Solving 3.5 + x = -3.640054945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -3.640054945 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -3.640054945 + -3.5 x = -3.640054945 + -3.5 Combine like terms: -3.640054945 + -3.5 = -7.140054945 x = -7.140054945 Simplifying x = -7.140054945Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.140054945, -7.140054945}Solution
x = {0, 0.140054945, -7.140054945}
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