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Simplifying 14x4 + -45x3 + 24x2 = 0 Reorder the terms: 24x2 + -45x3 + 14x4 = 0 Solving 24x2 + -45x3 + 14x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), 'x2'. x2(24 + -45x + 14x2) = 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 '(24 + -45x + 14x2)' equal to zero and attempt to solve: Simplifying 24 + -45x + 14x2 = 0 Solving 24 + -45x + 14x2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. 1.714285714 + -3.214285714x + x2 = 0 Move the constant term to the right: Add '-1.714285714' to each side of the equation. 1.714285714 + -3.214285714x + -1.714285714 + x2 = 0 + -1.714285714 Reorder the terms: 1.714285714 + -1.714285714 + -3.214285714x + x2 = 0 + -1.714285714 Combine like terms: 1.714285714 + -1.714285714 = 0.000000000 0.000000000 + -3.214285714x + x2 = 0 + -1.714285714 -3.214285714x + x2 = 0 + -1.714285714 Combine like terms: 0 + -1.714285714 = -1.714285714 -3.214285714x + x2 = -1.714285714 The x term is -3.214285714x. Take half its coefficient (-1.607142857). Square it (2.582908163) and add it to both sides. Add '2.582908163' to each side of the equation. -3.214285714x + 2.582908163 + x2 = -1.714285714 + 2.582908163 Reorder the terms: 2.582908163 + -3.214285714x + x2 = -1.714285714 + 2.582908163 Combine like terms: -1.714285714 + 2.582908163 = 0.868622449 2.582908163 + -3.214285714x + x2 = 0.868622449 Factor a perfect square on the left side: (x + -1.607142857)(x + -1.607142857) = 0.868622449 Calculate the square root of the right side: 0.931999168 Break this problem into two subproblems by setting (x + -1.607142857) equal to 0.931999168 and -0.931999168.Subproblem 1
x + -1.607142857 = 0.931999168 Simplifying x + -1.607142857 = 0.931999168 Reorder the terms: -1.607142857 + x = 0.931999168 Solving -1.607142857 + x = 0.931999168 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.607142857' to each side of the equation. -1.607142857 + 1.607142857 + x = 0.931999168 + 1.607142857 Combine like terms: -1.607142857 + 1.607142857 = 0.000000000 0.000000000 + x = 0.931999168 + 1.607142857 x = 0.931999168 + 1.607142857 Combine like terms: 0.931999168 + 1.607142857 = 2.539142025 x = 2.539142025 Simplifying x = 2.539142025Subproblem 2
x + -1.607142857 = -0.931999168 Simplifying x + -1.607142857 = -0.931999168 Reorder the terms: -1.607142857 + x = -0.931999168 Solving -1.607142857 + x = -0.931999168 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.607142857' to each side of the equation. -1.607142857 + 1.607142857 + x = -0.931999168 + 1.607142857 Combine like terms: -1.607142857 + 1.607142857 = 0.000000000 0.000000000 + x = -0.931999168 + 1.607142857 x = -0.931999168 + 1.607142857 Combine like terms: -0.931999168 + 1.607142857 = 0.675143689 x = 0.675143689 Simplifying x = 0.675143689Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.539142025, 0.675143689}Solution
x = {0, 2.539142025, 0.675143689}
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