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Simplifying 3x4 + -21x3 + 10x2 = 0 Reorder the terms: 10x2 + -21x3 + 3x4 = 0 Solving 10x2 + -21x3 + 3x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(10 + -21x + 3x2) = 0Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve: Simplifying x2 = 0 Solving x2 = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x2 = 0 Take the square root of each side: x = {0}Subproblem 2
Set the factor '(10 + -21x + 3x2)' equal to zero and attempt to solve: Simplifying 10 + -21x + 3x2 = 0 Solving 10 + -21x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 3.333333333 + -7x + x2 = 0 Move the constant term to the right: Add '-3.333333333' to each side of the equation. 3.333333333 + -7x + -3.333333333 + x2 = 0 + -3.333333333 Reorder the terms: 3.333333333 + -3.333333333 + -7x + x2 = 0 + -3.333333333 Combine like terms: 3.333333333 + -3.333333333 = 0.000000000 0.000000000 + -7x + x2 = 0 + -3.333333333 -7x + x2 = 0 + -3.333333333 Combine like terms: 0 + -3.333333333 = -3.333333333 -7x + x2 = -3.333333333 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 = -3.333333333 + 12.25 Reorder the terms: 12.25 + -7x + x2 = -3.333333333 + 12.25 Combine like terms: -3.333333333 + 12.25 = 8.916666667 12.25 + -7x + x2 = 8.916666667 Factor a perfect square on the left side: (x + -3.5)(x + -3.5) = 8.916666667 Calculate the square root of the right side: 2.986078811 Break this problem into two subproblems by setting (x + -3.5) equal to 2.986078811 and -2.986078811.Subproblem 1
x + -3.5 = 2.986078811 Simplifying x + -3.5 = 2.986078811 Reorder the terms: -3.5 + x = 2.986078811 Solving -3.5 + x = 2.986078811 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 = 2.986078811 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 2.986078811 + 3.5 x = 2.986078811 + 3.5 Combine like terms: 2.986078811 + 3.5 = 6.486078811 x = 6.486078811 Simplifying x = 6.486078811Subproblem 2
x + -3.5 = -2.986078811 Simplifying x + -3.5 = -2.986078811 Reorder the terms: -3.5 + x = -2.986078811 Solving -3.5 + x = -2.986078811 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 = -2.986078811 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -2.986078811 + 3.5 x = -2.986078811 + 3.5 Combine like terms: -2.986078811 + 3.5 = 0.513921189 x = 0.513921189 Simplifying x = 0.513921189Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.486078811, 0.513921189}Solution
x = {0, 6.486078811, 0.513921189}
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