-4x^2+21x=x+13

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Solution for -4x^2+21x=x+13 equation:


Simplifying
-4x2 + 21x = x + 13

Reorder the terms:
21x + -4x2 = x + 13

Reorder the terms:
21x + -4x2 = 13 + x

Solving
21x + -4x2 = 13 + x

Solving for variable 'x'.

Reorder the terms:
-13 + 21x + -1x + -4x2 = 13 + x + -13 + -1x

Combine like terms: 21x + -1x = 20x
-13 + 20x + -4x2 = 13 + x + -13 + -1x

Reorder the terms:
-13 + 20x + -4x2 = 13 + -13 + x + -1x

Combine like terms: 13 + -13 = 0
-13 + 20x + -4x2 = 0 + x + -1x
-13 + 20x + -4x2 = x + -1x

Combine like terms: x + -1x = 0
-13 + 20x + -4x2 = 0

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

Divide each side by '-4'.
3.25 + -5x + x2 = 0

Move the constant term to the right:

Add '-3.25' to each side of the equation.
3.25 + -5x + -3.25 + x2 = 0 + -3.25

Reorder the terms:
3.25 + -3.25 + -5x + x2 = 0 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + -5x + x2 = 0 + -3.25
-5x + x2 = 0 + -3.25

Combine like terms: 0 + -3.25 = -3.25
-5x + x2 = -3.25

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -3.25 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -3.25 + 6.25

Combine like terms: -3.25 + 6.25 = 3
6.25 + -5x + x2 = 3

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x + -2.5) equal to 1.732050808 and -1.732050808.

Subproblem 1

x + -2.5 = 1.732050808 Simplifying x + -2.5 = 1.732050808 Reorder the terms: -2.5 + x = 1.732050808 Solving -2.5 + x = 1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 1.732050808 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 1.732050808 + 2.5 x = 1.732050808 + 2.5 Combine like terms: 1.732050808 + 2.5 = 4.232050808 x = 4.232050808 Simplifying x = 4.232050808

Subproblem 2

x + -2.5 = -1.732050808 Simplifying x + -2.5 = -1.732050808 Reorder the terms: -2.5 + x = -1.732050808 Solving -2.5 + x = -1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -1.732050808 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -1.732050808 + 2.5 x = -1.732050808 + 2.5 Combine like terms: -1.732050808 + 2.5 = 0.767949192 x = 0.767949192 Simplifying x = 0.767949192

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.232050808, 0.767949192}

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