21X^4-21X^3-18X^2=0

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Solution for 21X^4-21X^3-18X^2=0 equation:


Simplifying
21X4 + -21X3 + -18X2 = 0

Reorder the terms:
-18X2 + -21X3 + 21X4 = 0

Solving
-18X2 + -21X3 + 21X4 = 0

Solving for variable 'X'.

Factor out the Greatest Common Factor (GCF), '3X2'.
3X2(-6 + -7X + 7X2) = 0

Ignore the factor 3.

Subproblem 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 + -7X + 7X2)' equal to zero and attempt to solve: Simplifying -6 + -7X + 7X2 = 0 Solving -6 + -7X + 7X2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.8571428571 + -1X + X2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + -1X + 0.8571428571 + X2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + -1X + X2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + -1X + X2 = 0 + 0.8571428571 -1X + X2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 -1X + X2 = 0.8571428571 The X term is -1X. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1X + 0.25 + X2 = 0.8571428571 + 0.25 Reorder the terms: 0.25 + -1X + X2 = 0.8571428571 + 0.25 Combine like terms: 0.8571428571 + 0.25 = 1.1071428571 0.25 + -1X + X2 = 1.1071428571 Factor a perfect square on the left side: (X + -0.5)(X + -0.5) = 1.1071428571 Calculate the square root of the right side: 1.052208562 Break this problem into two subproblems by setting (X + -0.5) equal to 1.052208562 and -1.052208562.

Subproblem 1

X + -0.5 = 1.052208562 Simplifying X + -0.5 = 1.052208562 Reorder the terms: -0.5 + X = 1.052208562 Solving -0.5 + X = 1.052208562 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + X = 1.052208562 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + X = 1.052208562 + 0.5 X = 1.052208562 + 0.5 Combine like terms: 1.052208562 + 0.5 = 1.552208562 X = 1.552208562 Simplifying X = 1.552208562

Subproblem 2

X + -0.5 = -1.052208562 Simplifying X + -0.5 = -1.052208562 Reorder the terms: -0.5 + X = -1.052208562 Solving -0.5 + X = -1.052208562 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + X = -1.052208562 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + X = -1.052208562 + 0.5 X = -1.052208562 + 0.5 Combine like terms: -1.052208562 + 0.5 = -0.552208562 X = -0.552208562 Simplifying X = -0.552208562

Solution

The solution to the problem is based on the solutions from the subproblems. X = {1.552208562, -0.552208562}

Solution

X = {0, 1.552208562, -0.552208562}

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