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Simplifying 6x4 + 5x3 = 120x2 Reorder the terms: 5x3 + 6x4 = 120x2 Solving 5x3 + 6x4 = 120x2 Solving for variable 'x'. Reorder the terms: -120x2 + 5x3 + 6x4 = 120x2 + -120x2 Combine like terms: 120x2 + -120x2 = 0 -120x2 + 5x3 + 6x4 = 0 Factor out the Greatest Common Factor (GCF), 'x2'. x2(-120 + 5x + 6x2) = 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 '(-120 + 5x + 6x2)' equal to zero and attempt to solve: Simplifying -120 + 5x + 6x2 = 0 Solving -120 + 5x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -20 + 0.8333333333x + x2 = 0 Move the constant term to the right: Add '20' to each side of the equation. -20 + 0.8333333333x + 20 + x2 = 0 + 20 Reorder the terms: -20 + 20 + 0.8333333333x + x2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + 0.8333333333x + x2 = 0 + 20 0.8333333333x + x2 = 0 + 20 Combine like terms: 0 + 20 = 20 0.8333333333x + x2 = 20 The x term is 0.8333333333x. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333x + 0.1736111111 + x2 = 20 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333x + x2 = 20 + 0.1736111111 Combine like terms: 20 + 0.1736111111 = 20.1736111111 0.1736111111 + 0.8333333333x + x2 = 20.1736111111 Factor a perfect square on the left side: (x + 0.4166666667)(x + 0.4166666667) = 20.1736111111 Calculate the square root of the right side: 4.491504326 Break this problem into two subproblems by setting (x + 0.4166666667) equal to 4.491504326 and -4.491504326.Subproblem 1
x + 0.4166666667 = 4.491504326 Simplifying x + 0.4166666667 = 4.491504326 Reorder the terms: 0.4166666667 + x = 4.491504326 Solving 0.4166666667 + x = 4.491504326 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = 4.491504326 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = 4.491504326 + -0.4166666667 x = 4.491504326 + -0.4166666667 Combine like terms: 4.491504326 + -0.4166666667 = 4.0748376593 x = 4.0748376593 Simplifying x = 4.0748376593Subproblem 2
x + 0.4166666667 = -4.491504326 Simplifying x + 0.4166666667 = -4.491504326 Reorder the terms: 0.4166666667 + x = -4.491504326 Solving 0.4166666667 + x = -4.491504326 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = -4.491504326 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = -4.491504326 + -0.4166666667 x = -4.491504326 + -0.4166666667 Combine like terms: -4.491504326 + -0.4166666667 = -4.9081709927 x = -4.9081709927 Simplifying x = -4.9081709927Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.0748376593, -4.9081709927}Solution
x = {0, 4.0748376593, -4.9081709927}
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