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Simplifying 0 = 10x4 + -24x3 + -3x2 Reorder the terms: 0 = -3x2 + -24x3 + 10x4 Solving 0 = -3x2 + -24x3 + 10x4 Solving for variable 'x'. Remove the zero: 3x2 + 24x3 + -10x4 = -3x2 + -24x3 + 10x4 + 3x2 + 24x3 + -10x4 Reorder the terms: 3x2 + 24x3 + -10x4 = -3x2 + 3x2 + -24x3 + 24x3 + 10x4 + -10x4 Combine like terms: -3x2 + 3x2 = 0 3x2 + 24x3 + -10x4 = 0 + -24x3 + 24x3 + 10x4 + -10x4 3x2 + 24x3 + -10x4 = -24x3 + 24x3 + 10x4 + -10x4 Combine like terms: -24x3 + 24x3 = 0 3x2 + 24x3 + -10x4 = 0 + 10x4 + -10x4 3x2 + 24x3 + -10x4 = 10x4 + -10x4 Combine like terms: 10x4 + -10x4 = 0 3x2 + 24x3 + -10x4 = 0 Factor out the Greatest Common Factor (GCF), 'x2'. x2(3 + 24x + -10x2) = 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 '(3 + 24x + -10x2)' equal to zero and attempt to solve: Simplifying 3 + 24x + -10x2 = 0 Solving 3 + 24x + -10x2 = 0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -0.3 + -2.4x + x2 = 0 Move the constant term to the right: Add '0.3' to each side of the equation. -0.3 + -2.4x + 0.3 + x2 = 0 + 0.3 Reorder the terms: -0.3 + 0.3 + -2.4x + x2 = 0 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + -2.4x + x2 = 0 + 0.3 -2.4x + x2 = 0 + 0.3 Combine like terms: 0 + 0.3 = 0.3 -2.4x + x2 = 0.3 The x term is -2.4x. Take half its coefficient (-1.2). Square it (1.44) and add it to both sides. Add '1.44' to each side of the equation. -2.4x + 1.44 + x2 = 0.3 + 1.44 Reorder the terms: 1.44 + -2.4x + x2 = 0.3 + 1.44 Combine like terms: 0.3 + 1.44 = 1.74 1.44 + -2.4x + x2 = 1.74 Factor a perfect square on the left side: (x + -1.2)(x + -1.2) = 1.74 Calculate the square root of the right side: 1.319090596 Break this problem into two subproblems by setting (x + -1.2) equal to 1.319090596 and -1.319090596.Subproblem 1
x + -1.2 = 1.319090596 Simplifying x + -1.2 = 1.319090596 Reorder the terms: -1.2 + x = 1.319090596 Solving -1.2 + x = 1.319090596 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + x = 1.319090596 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + x = 1.319090596 + 1.2 x = 1.319090596 + 1.2 Combine like terms: 1.319090596 + 1.2 = 2.519090596 x = 2.519090596 Simplifying x = 2.519090596Subproblem 2
x + -1.2 = -1.319090596 Simplifying x + -1.2 = -1.319090596 Reorder the terms: -1.2 + x = -1.319090596 Solving -1.2 + x = -1.319090596 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.2' to each side of the equation. -1.2 + 1.2 + x = -1.319090596 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + x = -1.319090596 + 1.2 x = -1.319090596 + 1.2 Combine like terms: -1.319090596 + 1.2 = -0.119090596 x = -0.119090596 Simplifying x = -0.119090596Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.519090596, -0.119090596}Solution
x = {0, 2.519090596, -0.119090596}
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