6x^2+21x-5=0

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Solution for 6x^2+21x-5=0 equation:


Simplifying
6x2 + 21x + -5 = 0

Reorder the terms:
-5 + 21x + 6x2 = 0

Solving
-5 + 21x + 6x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
6 the coefficient of the squared term: 

Divide each side by '6'.
-0.8333333333 + 3.5x + x2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + 3.5x + 0.8333333333 + x2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + 3.5x + x2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + 3.5x + x2 = 0 + 0.8333333333
3.5x + x2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
3.5x + x2 = 0.8333333333

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = 0.8333333333 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = 0.8333333333 + 3.0625

Combine like terms: 0.8333333333 + 3.0625 = 3.8958333333
3.0625 + 3.5x + x2 = 3.8958333333

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 3.8958333333

Calculate the square root of the right side: 1.973786547

Break this problem into two subproblems by setting 
(x + 1.75) equal to 1.973786547 and -1.973786547.

Subproblem 1

x + 1.75 = 1.973786547 Simplifying x + 1.75 = 1.973786547 Reorder the terms: 1.75 + x = 1.973786547 Solving 1.75 + x = 1.973786547 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 1.973786547 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 1.973786547 + -1.75 x = 1.973786547 + -1.75 Combine like terms: 1.973786547 + -1.75 = 0.223786547 x = 0.223786547 Simplifying x = 0.223786547

Subproblem 2

x + 1.75 = -1.973786547 Simplifying x + 1.75 = -1.973786547 Reorder the terms: 1.75 + x = -1.973786547 Solving 1.75 + x = -1.973786547 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -1.973786547 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -1.973786547 + -1.75 x = -1.973786547 + -1.75 Combine like terms: -1.973786547 + -1.75 = -3.723786547 x = -3.723786547 Simplifying x = -3.723786547

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.223786547, -3.723786547}

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