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Simplifying 6x3 + 32x2 = 63x Reorder the terms: 32x2 + 6x3 = 63x Solving 32x2 + 6x3 = 63x Solving for variable 'x'. Reorder the terms: -63x + 32x2 + 6x3 = 63x + -63x Combine like terms: 63x + -63x = 0 -63x + 32x2 + 6x3 = 0 Factor out the Greatest Common Factor (GCF), 'x'. x(-63 + 32x + 6x2) = 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 '(-63 + 32x + 6x2)' equal to zero and attempt to solve: Simplifying -63 + 32x + 6x2 = 0 Solving -63 + 32x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -10.5 + 5.333333333x + x2 = 0 Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + 5.333333333x + 10.5 + x2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + 5.333333333x + x2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + 5.333333333x + x2 = 0 + 10.5 5.333333333x + x2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 5.333333333x + x2 = 10.5 The x term is 5.333333333x. Take half its coefficient (2.666666667). Square it (7.111111113) and add it to both sides. Add '7.111111113' to each side of the equation. 5.333333333x + 7.111111113 + x2 = 10.5 + 7.111111113 Reorder the terms: 7.111111113 + 5.333333333x + x2 = 10.5 + 7.111111113 Combine like terms: 10.5 + 7.111111113 = 17.611111113 7.111111113 + 5.333333333x + x2 = 17.611111113 Factor a perfect square on the left side: (x + 2.666666667)(x + 2.666666667) = 17.611111113 Calculate the square root of the right side: 4.196559438 Break this problem into two subproblems by setting (x + 2.666666667) equal to 4.196559438 and -4.196559438.Subproblem 1
x + 2.666666667 = 4.196559438 Simplifying x + 2.666666667 = 4.196559438 Reorder the terms: 2.666666667 + x = 4.196559438 Solving 2.666666667 + x = 4.196559438 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + x = 4.196559438 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + x = 4.196559438 + -2.666666667 x = 4.196559438 + -2.666666667 Combine like terms: 4.196559438 + -2.666666667 = 1.529892771 x = 1.529892771 Simplifying x = 1.529892771Subproblem 2
x + 2.666666667 = -4.196559438 Simplifying x + 2.666666667 = -4.196559438 Reorder the terms: 2.666666667 + x = -4.196559438 Solving 2.666666667 + x = -4.196559438 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + x = -4.196559438 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + x = -4.196559438 + -2.666666667 x = -4.196559438 + -2.666666667 Combine like terms: -4.196559438 + -2.666666667 = -6.863226105 x = -6.863226105 Simplifying x = -6.863226105Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.529892771, -6.863226105}Solution
x = {0, 1.529892771, -6.863226105}
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