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Simplifying (20 + -2x)(40 + -2x) * x = x Reorder the terms for easier multiplication: x(20 + -2x)(40 + -2x) = x Multiply (20 + -2x) * (40 + -2x) x(20(40 + -2x) + -2x * (40 + -2x)) = x x((40 * 20 + -2x * 20) + -2x * (40 + -2x)) = x x((800 + -40x) + -2x * (40 + -2x)) = x x(800 + -40x + (40 * -2x + -2x * -2x)) = x x(800 + -40x + (-80x + 4x2)) = x Combine like terms: -40x + -80x = -120x x(800 + -120x + 4x2) = x (800 * x + -120x * x + 4x2 * x) = x (800x + -120x2 + 4x3) = x Solving 800x + -120x2 + 4x3 = x Solving for variable 'x'. Reorder the terms: 800x + -1x + -120x2 + 4x3 = x + -1x Combine like terms: 800x + -1x = 799x 799x + -120x2 + 4x3 = x + -1x Combine like terms: x + -1x = 0 799x + -120x2 + 4x3 = 0 Factor out the Greatest Common Factor (GCF), 'x'. x(799 + -120x + 4x2) = 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 '(799 + -120x + 4x2)' equal to zero and attempt to solve: Simplifying 799 + -120x + 4x2 = 0 Solving 799 + -120x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 199.75 + -30x + x2 = 0 Move the constant term to the right: Add '-199.75' to each side of the equation. 199.75 + -30x + -199.75 + x2 = 0 + -199.75 Reorder the terms: 199.75 + -199.75 + -30x + x2 = 0 + -199.75 Combine like terms: 199.75 + -199.75 = 0.00 0.00 + -30x + x2 = 0 + -199.75 -30x + x2 = 0 + -199.75 Combine like terms: 0 + -199.75 = -199.75 -30x + x2 = -199.75 The x term is -30x. Take half its coefficient (-15). Square it (225) and add it to both sides. Add '225' to each side of the equation. -30x + 225 + x2 = -199.75 + 225 Reorder the terms: 225 + -30x + x2 = -199.75 + 225 Combine like terms: -199.75 + 225 = 25.25 225 + -30x + x2 = 25.25 Factor a perfect square on the left side: (x + -15)(x + -15) = 25.25 Calculate the square root of the right side: 5.024937811 Break this problem into two subproblems by setting (x + -15) equal to 5.024937811 and -5.024937811.Subproblem 1
x + -15 = 5.024937811 Simplifying x + -15 = 5.024937811 Reorder the terms: -15 + x = 5.024937811 Solving -15 + x = 5.024937811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '15' to each side of the equation. -15 + 15 + x = 5.024937811 + 15 Combine like terms: -15 + 15 = 0 0 + x = 5.024937811 + 15 x = 5.024937811 + 15 Combine like terms: 5.024937811 + 15 = 20.024937811 x = 20.024937811 Simplifying x = 20.024937811Subproblem 2
x + -15 = -5.024937811 Simplifying x + -15 = -5.024937811 Reorder the terms: -15 + x = -5.024937811 Solving -15 + x = -5.024937811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '15' to each side of the equation. -15 + 15 + x = -5.024937811 + 15 Combine like terms: -15 + 15 = 0 0 + x = -5.024937811 + 15 x = -5.024937811 + 15 Combine like terms: -5.024937811 + 15 = 9.975062189 x = 9.975062189 Simplifying x = 9.975062189Solution
The solution to the problem is based on the solutions from the subproblems. x = {20.024937811, 9.975062189}Solution
x = {0, 20.024937811, 9.975062189}
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