4x+2x^2=53

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Solution for 4x+2x^2=53 equation:


Simplifying
4x + 2x2 = 53

Solving
4x + 2x2 = 53

Solving for variable 'x'.

Reorder the terms:
-53 + 4x + 2x2 = 53 + -53

Combine like terms: 53 + -53 = 0
-53 + 4x + 2x2 = 0

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

Divide each side by '2'.
-26.5 + 2x + x2 = 0

Move the constant term to the right:

Add '26.5' to each side of the equation.
-26.5 + 2x + 26.5 + x2 = 0 + 26.5

Reorder the terms:
-26.5 + 26.5 + 2x + x2 = 0 + 26.5

Combine like terms: -26.5 + 26.5 = 0.0
0.0 + 2x + x2 = 0 + 26.5
2x + x2 = 0 + 26.5

Combine like terms: 0 + 26.5 = 26.5
2x + x2 = 26.5

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 26.5 + 1

Reorder the terms:
1 + 2x + x2 = 26.5 + 1

Combine like terms: 26.5 + 1 = 27.5
1 + 2x + x2 = 27.5

Factor a perfect square on the left side:
(x + 1)(x + 1) = 27.5

Calculate the square root of the right side: 5.244044241

Break this problem into two subproblems by setting 
(x + 1) equal to 5.244044241 and -5.244044241.

Subproblem 1

x + 1 = 5.244044241 Simplifying x + 1 = 5.244044241 Reorder the terms: 1 + x = 5.244044241 Solving 1 + x = 5.244044241 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 5.244044241 + -1 Combine like terms: 1 + -1 = 0 0 + x = 5.244044241 + -1 x = 5.244044241 + -1 Combine like terms: 5.244044241 + -1 = 4.244044241 x = 4.244044241 Simplifying x = 4.244044241

Subproblem 2

x + 1 = -5.244044241 Simplifying x + 1 = -5.244044241 Reorder the terms: 1 + x = -5.244044241 Solving 1 + x = -5.244044241 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -5.244044241 + -1 Combine like terms: 1 + -1 = 0 0 + x = -5.244044241 + -1 x = -5.244044241 + -1 Combine like terms: -5.244044241 + -1 = -6.244044241 x = -6.244044241 Simplifying x = -6.244044241

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.244044241, -6.244044241}

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