4x^2=4x+7

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Solution for 4x^2=4x+7 equation:


Simplifying
4x2 = 4x + 7

Reorder the terms:
4x2 = 7 + 4x

Solving
4x2 = 7 + 4x

Solving for variable 'x'.

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

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

Combine like terms: 7 + -7 = 0
-7 + -4x + 4x2 = 0 + 4x + -4x
-7 + -4x + 4x2 = 4x + -4x

Combine like terms: 4x + -4x = 0
-7 + -4x + 4x2 = 0

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

Divide each side by '4'.
-1.75 + -1x + x2 = 0

Move the constant term to the right:

Add '1.75' to each side of the equation.
-1.75 + -1x + 1.75 + x2 = 0 + 1.75

Reorder the terms:
-1.75 + 1.75 + -1x + x2 = 0 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + -1x + x2 = 0 + 1.75
-1x + x2 = 0 + 1.75

Combine like terms: 0 + 1.75 = 1.75
-1x + x2 = 1.75

The x term is -1x.  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.
-1x + 0.25 + x2 = 1.75 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 1.75 + 0.25

Combine like terms: 1.75 + 0.25 = 2
0.25 + -1x + x2 = 2

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + -0.5) equal to 1.414213562 and -1.414213562.

Subproblem 1

x + -0.5 = 1.414213562 Simplifying x + -0.5 = 1.414213562 Reorder the terms: -0.5 + x = 1.414213562 Solving -0.5 + x = 1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 1.414213562 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 1.414213562 + 0.5 x = 1.414213562 + 0.5 Combine like terms: 1.414213562 + 0.5 = 1.914213562 x = 1.914213562 Simplifying x = 1.914213562

Subproblem 2

x + -0.5 = -1.414213562 Simplifying x + -0.5 = -1.414213562 Reorder the terms: -0.5 + x = -1.414213562 Solving -0.5 + x = -1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -1.414213562 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -1.414213562 + 0.5 x = -1.414213562 + 0.5 Combine like terms: -1.414213562 + 0.5 = -0.914213562 x = -0.914213562 Simplifying x = -0.914213562

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.914213562, -0.914213562}

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