0=x^4+7x^3-3x^2

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Solution for 0=x^4+7x^3-3x^2 equation:


Simplifying
0 = x4 + 7x3 + -3x2

Reorder the terms:
0 = -3x2 + 7x3 + x4

Solving
0 = -3x2 + 7x3 + x4

Solving for variable 'x'.
Remove the zero:
3x2 + -7x3 + -1x4 = -3x2 + 7x3 + x4 + 3x2 + -7x3 + -1x4

Reorder the terms:
3x2 + -7x3 + -1x4 = -3x2 + 3x2 + 7x3 + -7x3 + x4 + -1x4

Combine like terms: -3x2 + 3x2 = 0
3x2 + -7x3 + -1x4 = 0 + 7x3 + -7x3 + x4 + -1x4
3x2 + -7x3 + -1x4 = 7x3 + -7x3 + x4 + -1x4

Combine like terms: 7x3 + -7x3 = 0
3x2 + -7x3 + -1x4 = 0 + x4 + -1x4
3x2 + -7x3 + -1x4 = x4 + -1x4

Combine like terms: x4 + -1x4 = 0
3x2 + -7x3 + -1x4 = 0

Factor out the Greatest Common Factor (GCF), 'x2'.
x2(3 + -7x + -1x2) = 0

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 '(3 + -7x + -1x2)' equal to zero and attempt to solve: Simplifying 3 + -7x + -1x2 = 0 Solving 3 + -7x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -3 + 7x + x2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 7x + 3 + x2 = 0 + 3 Reorder the terms: -3 + 3 + 7x + x2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 7x + x2 = 0 + 3 7x + x2 = 0 + 3 Combine like terms: 0 + 3 = 3 7x + x2 = 3 The x term is 7x. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7x + 12.25 + x2 = 3 + 12.25 Reorder the terms: 12.25 + 7x + x2 = 3 + 12.25 Combine like terms: 3 + 12.25 = 15.25 12.25 + 7x + x2 = 15.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 15.25 Calculate the square root of the right side: 3.905124838 Break this problem into two subproblems by setting (x + 3.5) equal to 3.905124838 and -3.905124838.

Subproblem 1

x + 3.5 = 3.905124838 Simplifying x + 3.5 = 3.905124838 Reorder the terms: 3.5 + x = 3.905124838 Solving 3.5 + x = 3.905124838 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 3.905124838 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 3.905124838 + -3.5 x = 3.905124838 + -3.5 Combine like terms: 3.905124838 + -3.5 = 0.405124838 x = 0.405124838 Simplifying x = 0.405124838

Subproblem 2

x + 3.5 = -3.905124838 Simplifying x + 3.5 = -3.905124838 Reorder the terms: 3.5 + x = -3.905124838 Solving 3.5 + x = -3.905124838 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -3.905124838 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -3.905124838 + -3.5 x = -3.905124838 + -3.5 Combine like terms: -3.905124838 + -3.5 = -7.405124838 x = -7.405124838 Simplifying x = -7.405124838

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.405124838, -7.405124838}

Solution

x = {0, 0.405124838, -7.405124838}

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