4x^2+6x=21

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


Simplifying
4x2 + 6x = 21

Reorder the terms:
6x + 4x2 = 21

Solving
6x + 4x2 = 21

Solving for variable 'x'.

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

Combine like terms: 21 + -21 = 0
-21 + 6x + 4x2 = 0

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

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

Move the constant term to the right:

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

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

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + 1.5x + x2 = 0 + 5.25
1.5x + x2 = 0 + 5.25

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

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 5.25 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 5.25 + 0.5625

Combine like terms: 5.25 + 0.5625 = 5.8125
0.5625 + 1.5x + x2 = 5.8125

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 5.8125

Calculate the square root of the right side: 2.41091269

Break this problem into two subproblems by setting 
(x + 0.75) equal to 2.41091269 and -2.41091269.

Subproblem 1

x + 0.75 = 2.41091269 Simplifying x + 0.75 = 2.41091269 Reorder the terms: 0.75 + x = 2.41091269 Solving 0.75 + x = 2.41091269 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 2.41091269 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 2.41091269 + -0.75 x = 2.41091269 + -0.75 Combine like terms: 2.41091269 + -0.75 = 1.66091269 x = 1.66091269 Simplifying x = 1.66091269

Subproblem 2

x + 0.75 = -2.41091269 Simplifying x + 0.75 = -2.41091269 Reorder the terms: 0.75 + x = -2.41091269 Solving 0.75 + x = -2.41091269 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -2.41091269 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -2.41091269 + -0.75 x = -2.41091269 + -0.75 Combine like terms: -2.41091269 + -0.75 = -3.16091269 x = -3.16091269 Simplifying x = -3.16091269

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.66091269, -3.16091269}

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