6x+7=4x^2-14

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Solution for 6x+7=4x^2-14 equation:


Simplifying
6x + 7 = 4x2 + -14

Reorder the terms:
7 + 6x = 4x2 + -14

Reorder the terms:
7 + 6x = -14 + 4x2

Solving
7 + 6x = -14 + 4x2

Solving for variable 'x'.

Reorder the terms:
7 + 14 + 6x + -4x2 = -14 + 4x2 + 14 + -4x2

Combine like terms: 7 + 14 = 21
21 + 6x + -4x2 = -14 + 4x2 + 14 + -4x2

Reorder the terms:
21 + 6x + -4x2 = -14 + 14 + 4x2 + -4x2

Combine like terms: -14 + 14 = 0
21 + 6x + -4x2 = 0 + 4x2 + -4x2
21 + 6x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 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 = 3.16091269 x = 3.16091269 Simplifying x = 3.16091269

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 = -1.66091269 x = -1.66091269 Simplifying x = -1.66091269

Solution

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

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