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Simplifying 7x4 + 10x3 + -6x2 = 0 Reorder the terms: -6x2 + 10x3 + 7x4 = 0 Solving -6x2 + 10x3 + 7x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(-6 + 10x + 7x2) = 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 '(-6 + 10x + 7x2)' equal to zero and attempt to solve: Simplifying -6 + 10x + 7x2 = 0 Solving -6 + 10x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.8571428571 + 1.428571429x + x2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + 1.428571429x + 0.8571428571 + x2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + 1.428571429x + x2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + 1.428571429x + x2 = 0 + 0.8571428571 1.428571429x + x2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 1.428571429x + x2 = 0.8571428571 The x term is 1.428571429x. Take half its coefficient (0.7142857145). Square it (0.5102040819) and add it to both sides. Add '0.5102040819' to each side of the equation. 1.428571429x + 0.5102040819 + x2 = 0.8571428571 + 0.5102040819 Reorder the terms: 0.5102040819 + 1.428571429x + x2 = 0.8571428571 + 0.5102040819 Combine like terms: 0.8571428571 + 0.5102040819 = 1.367346939 0.5102040819 + 1.428571429x + x2 = 1.367346939 Factor a perfect square on the left side: (x + 0.7142857145)(x + 0.7142857145) = 1.367346939 Calculate the square root of the right side: 1.16933611 Break this problem into two subproblems by setting (x + 0.7142857145) equal to 1.16933611 and -1.16933611.Subproblem 1
x + 0.7142857145 = 1.16933611 Simplifying x + 0.7142857145 = 1.16933611 Reorder the terms: 0.7142857145 + x = 1.16933611 Solving 0.7142857145 + x = 1.16933611 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7142857145' to each side of the equation. 0.7142857145 + -0.7142857145 + x = 1.16933611 + -0.7142857145 Combine like terms: 0.7142857145 + -0.7142857145 = 0.0000000000 0.0000000000 + x = 1.16933611 + -0.7142857145 x = 1.16933611 + -0.7142857145 Combine like terms: 1.16933611 + -0.7142857145 = 0.4550503955 x = 0.4550503955 Simplifying x = 0.4550503955Subproblem 2
x + 0.7142857145 = -1.16933611 Simplifying x + 0.7142857145 = -1.16933611 Reorder the terms: 0.7142857145 + x = -1.16933611 Solving 0.7142857145 + x = -1.16933611 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7142857145' to each side of the equation. 0.7142857145 + -0.7142857145 + x = -1.16933611 + -0.7142857145 Combine like terms: 0.7142857145 + -0.7142857145 = 0.0000000000 0.0000000000 + x = -1.16933611 + -0.7142857145 x = -1.16933611 + -0.7142857145 Combine like terms: -1.16933611 + -0.7142857145 = -1.8836218245 x = -1.8836218245 Simplifying x = -1.8836218245Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.4550503955, -1.8836218245}Solution
x = {0, 0.4550503955, -1.8836218245}
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