2x^2=x+20

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Solution for 2x^2=x+20 equation:


Simplifying
2x2 = x + 20

Reorder the terms:
2x2 = 20 + x

Solving
2x2 = 20 + x

Solving for variable 'x'.

Reorder the terms:
-20 + -1x + 2x2 = 20 + x + -20 + -1x

Reorder the terms:
-20 + -1x + 2x2 = 20 + -20 + x + -1x

Combine like terms: 20 + -20 = 0
-20 + -1x + 2x2 = 0 + x + -1x
-20 + -1x + 2x2 = x + -1x

Combine like terms: x + -1x = 0
-20 + -1x + 2x2 = 0

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

Divide each side by '2'.
-10 + -0.5x + x2 = 0

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -0.5x + 10 + x2 = 0 + 10

Reorder the terms:
-10 + 10 + -0.5x + x2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -0.5x + x2 = 0 + 10
-0.5x + x2 = 0 + 10

Combine like terms: 0 + 10 = 10
-0.5x + x2 = 10

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 = 10 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 10 + 0.0625

Combine like terms: 10 + 0.0625 = 10.0625
0.0625 + -0.5x + x2 = 10.0625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 10.0625

Calculate the square root of the right side: 3.172144385

Break this problem into two subproblems by setting 
(x + -0.25) equal to 3.172144385 and -3.172144385.

Subproblem 1

x + -0.25 = 3.172144385 Simplifying x + -0.25 = 3.172144385 Reorder the terms: -0.25 + x = 3.172144385 Solving -0.25 + x = 3.172144385 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 = 3.172144385 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 3.172144385 + 0.25 x = 3.172144385 + 0.25 Combine like terms: 3.172144385 + 0.25 = 3.422144385 x = 3.422144385 Simplifying x = 3.422144385

Subproblem 2

x + -0.25 = -3.172144385 Simplifying x + -0.25 = -3.172144385 Reorder the terms: -0.25 + x = -3.172144385 Solving -0.25 + x = -3.172144385 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 = -3.172144385 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -3.172144385 + 0.25 x = -3.172144385 + 0.25 Combine like terms: -3.172144385 + 0.25 = -2.922144385 x = -2.922144385 Simplifying x = -2.922144385

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.422144385, -2.922144385}

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