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Simplifying (4x + 16) * x(3 + 2x) = 10x Reorder the terms: (16 + 4x) * x(3 + 2x) = 10x Reorder the terms for easier multiplication: x(16 + 4x)(3 + 2x) = 10x Multiply (16 + 4x) * (3 + 2x) x(16(3 + 2x) + 4x * (3 + 2x)) = 10x x((3 * 16 + 2x * 16) + 4x * (3 + 2x)) = 10x x((48 + 32x) + 4x * (3 + 2x)) = 10x x(48 + 32x + (3 * 4x + 2x * 4x)) = 10x x(48 + 32x + (12x + 8x2)) = 10x Combine like terms: 32x + 12x = 44x x(48 + 44x + 8x2) = 10x (48 * x + 44x * x + 8x2 * x) = 10x (48x + 44x2 + 8x3) = 10x Solving 48x + 44x2 + 8x3 = 10x Solving for variable 'x'. Reorder the terms: 48x + -10x + 44x2 + 8x3 = 10x + -10x Combine like terms: 48x + -10x = 38x 38x + 44x2 + 8x3 = 10x + -10x Combine like terms: 10x + -10x = 0 38x + 44x2 + 8x3 = 0 Factor out the Greatest Common Factor (GCF), '2x'. 2x(19 + 22x + 4x2) = 0 Ignore the factor 2.Subproblem 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 '(19 + 22x + 4x2)' equal to zero and attempt to solve: Simplifying 19 + 22x + 4x2 = 0 Solving 19 + 22x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 4.75 + 5.5x + x2 = 0 Move the constant term to the right: Add '-4.75' to each side of the equation. 4.75 + 5.5x + -4.75 + x2 = 0 + -4.75 Reorder the terms: 4.75 + -4.75 + 5.5x + x2 = 0 + -4.75 Combine like terms: 4.75 + -4.75 = 0.00 0.00 + 5.5x + x2 = 0 + -4.75 5.5x + x2 = 0 + -4.75 Combine like terms: 0 + -4.75 = -4.75 5.5x + x2 = -4.75 The x term is 5.5x. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5x + 7.5625 + x2 = -4.75 + 7.5625 Reorder the terms: 7.5625 + 5.5x + x2 = -4.75 + 7.5625 Combine like terms: -4.75 + 7.5625 = 2.8125 7.5625 + 5.5x + x2 = 2.8125 Factor a perfect square on the left side: (x + 2.75)(x + 2.75) = 2.8125 Calculate the square root of the right side: 1.677050983 Break this problem into two subproblems by setting (x + 2.75) equal to 1.677050983 and -1.677050983.Subproblem 1
x + 2.75 = 1.677050983 Simplifying x + 2.75 = 1.677050983 Reorder the terms: 2.75 + x = 1.677050983 Solving 2.75 + x = 1.677050983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = 1.677050983 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = 1.677050983 + -2.75 x = 1.677050983 + -2.75 Combine like terms: 1.677050983 + -2.75 = -1.072949017 x = -1.072949017 Simplifying x = -1.072949017Subproblem 2
x + 2.75 = -1.677050983 Simplifying x + 2.75 = -1.677050983 Reorder the terms: 2.75 + x = -1.677050983 Solving 2.75 + x = -1.677050983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = -1.677050983 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = -1.677050983 + -2.75 x = -1.677050983 + -2.75 Combine like terms: -1.677050983 + -2.75 = -4.427050983 x = -4.427050983 Simplifying x = -4.427050983Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.072949017, -4.427050983}Solution
x = {0, -1.072949017, -4.427050983}
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