4x^2+21x-6=0

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Solution for 4x^2+21x-6=0 equation:


Simplifying
4x2 + 21x + -6 = 0

Reorder the terms:
-6 + 21x + 4x2 = 0

Solving
-6 + 21x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-1.5 + 5.25x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 5.25x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 5.25x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 5.25x + x2 = 0 + 1.5
5.25x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
5.25x + x2 = 1.5

The x term is 5.25x.  Take half its coefficient (2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
5.25x + 6.890625 + x2 = 1.5 + 6.890625

Reorder the terms:
6.890625 + 5.25x + x2 = 1.5 + 6.890625

Combine like terms: 1.5 + 6.890625 = 8.390625
6.890625 + 5.25x + x2 = 8.390625

Factor a perfect square on the left side:
(x + 2.625)(x + 2.625) = 8.390625

Calculate the square root of the right side: 2.896657557

Break this problem into two subproblems by setting 
(x + 2.625) equal to 2.896657557 and -2.896657557.

Subproblem 1

x + 2.625 = 2.896657557 Simplifying x + 2.625 = 2.896657557 Reorder the terms: 2.625 + x = 2.896657557 Solving 2.625 + x = 2.896657557 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.625' to each side of the equation. 2.625 + -2.625 + x = 2.896657557 + -2.625 Combine like terms: 2.625 + -2.625 = 0.000 0.000 + x = 2.896657557 + -2.625 x = 2.896657557 + -2.625 Combine like terms: 2.896657557 + -2.625 = 0.271657557 x = 0.271657557 Simplifying x = 0.271657557

Subproblem 2

x + 2.625 = -2.896657557 Simplifying x + 2.625 = -2.896657557 Reorder the terms: 2.625 + x = -2.896657557 Solving 2.625 + x = -2.896657557 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.625' to each side of the equation. 2.625 + -2.625 + x = -2.896657557 + -2.625 Combine like terms: 2.625 + -2.625 = 0.000 0.000 + x = -2.896657557 + -2.625 x = -2.896657557 + -2.625 Combine like terms: -2.896657557 + -2.625 = -5.521657557 x = -5.521657557 Simplifying x = -5.521657557

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.271657557, -5.521657557}

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