4x^2=2x+7

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


Simplifying
4x2 = 2x + 7

Reorder the terms:
4x2 = 7 + 2x

Solving
4x2 = 7 + 2x

Solving for variable 'x'.

Reorder the terms:
-7 + -2x + 4x2 = 7 + 2x + -7 + -2x

Reorder the terms:
-7 + -2x + 4x2 = 7 + -7 + 2x + -2x

Combine like terms: 7 + -7 = 0
-7 + -2x + 4x2 = 0 + 2x + -2x
-7 + -2x + 4x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-7 + -2x + 4x2 = 0

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

Divide each side by '4'.
-1.75 + -0.5x + x2 = 0

Move the constant term to the right:

Add '1.75' to each side of the equation.
-1.75 + -0.5x + 1.75 + x2 = 0 + 1.75

Reorder the terms:
-1.75 + 1.75 + -0.5x + x2 = 0 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + -0.5x + x2 = 0 + 1.75
-0.5x + x2 = 0 + 1.75

Combine like terms: 0 + 1.75 = 1.75
-0.5x + x2 = 1.75

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 1.75 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 1.75 + 0.0625

Combine like terms: 1.75 + 0.0625 = 1.8125
0.0625 + -0.5x + x2 = 1.8125

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 1.8125

Calculate the square root of the right side: 1.346291202

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.346291202 and -1.346291202.

Subproblem 1

x + -0.25 = 1.346291202 Simplifying x + -0.25 = 1.346291202 Reorder the terms: -0.25 + x = 1.346291202 Solving -0.25 + x = 1.346291202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 1.346291202 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 1.346291202 + 0.25 x = 1.346291202 + 0.25 Combine like terms: 1.346291202 + 0.25 = 1.596291202 x = 1.596291202 Simplifying x = 1.596291202

Subproblem 2

x + -0.25 = -1.346291202 Simplifying x + -0.25 = -1.346291202 Reorder the terms: -0.25 + x = -1.346291202 Solving -0.25 + x = -1.346291202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -1.346291202 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -1.346291202 + 0.25 x = -1.346291202 + 0.25 Combine like terms: -1.346291202 + 0.25 = -1.096291202 x = -1.096291202 Simplifying x = -1.096291202

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.596291202, -1.096291202}

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