(6c^2+5c-3)-(3c^2+8c)=0

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Solution for (6c^2+5c-3)-(3c^2+8c)=0 equation:


Simplifying
(6c2 + 5c + -3) + -1(3c2 + 8c) = 0

Reorder the terms:
(-3 + 5c + 6c2) + -1(3c2 + 8c) = 0

Remove parenthesis around (-3 + 5c + 6c2)
-3 + 5c + 6c2 + -1(3c2 + 8c) = 0

Reorder the terms:
-3 + 5c + 6c2 + -1(8c + 3c2) = 0
-3 + 5c + 6c2 + (8c * -1 + 3c2 * -1) = 0
-3 + 5c + 6c2 + (-8c + -3c2) = 0

Reorder the terms:
-3 + 5c + -8c + 6c2 + -3c2 = 0

Combine like terms: 5c + -8c = -3c
-3 + -3c + 6c2 + -3c2 = 0

Combine like terms: 6c2 + -3c2 = 3c2
-3 + -3c + 3c2 = 0

Solving
-3 + -3c + 3c2 = 0

Solving for variable 'c'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-1 + -1c + c2) = 0

Ignore the factor 3.

Subproblem 1

Set the factor '(-1 + -1c + c2)' equal to zero and attempt to solve: Simplifying -1 + -1c + c2 = 0 Solving -1 + -1c + c2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -1c + 1 + c2 = 0 + 1 Reorder the terms: -1 + 1 + -1c + c2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1c + c2 = 0 + 1 -1c + c2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1c + c2 = 1 The c term is -1c. 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. -1c + 0.25 + c2 = 1 + 0.25 Reorder the terms: 0.25 + -1c + c2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + -1c + c2 = 1.25 Factor a perfect square on the left side: (c + -0.5)(c + -0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (c + -0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

c + -0.5 = 1.118033989 Simplifying c + -0.5 = 1.118033989 Reorder the terms: -0.5 + c = 1.118033989 Solving -0.5 + c = 1.118033989 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + c = 1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + c = 1.118033989 + 0.5 c = 1.118033989 + 0.5 Combine like terms: 1.118033989 + 0.5 = 1.618033989 c = 1.618033989 Simplifying c = 1.618033989

Subproblem 2

c + -0.5 = -1.118033989 Simplifying c + -0.5 = -1.118033989 Reorder the terms: -0.5 + c = -1.118033989 Solving -0.5 + c = -1.118033989 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + c = -1.118033989 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + c = -1.118033989 + 0.5 c = -1.118033989 + 0.5 Combine like terms: -1.118033989 + 0.5 = -0.618033989 c = -0.618033989 Simplifying c = -0.618033989

Solution

The solution to the problem is based on the solutions from the subproblems. c = {1.618033989, -0.618033989}

Solution

c = {1.618033989, -0.618033989}

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