4x^2-11x+24=5x+10

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Solution for 4x^2-11x+24=5x+10 equation: 


Simplifying
4x2 + -11x + 24 = 5x + 10

Reorder the terms:
24 + -11x + 4x2 = 5x + 10

Reorder the terms:
24 + -11x + 4x2 = 10 + 5x

Solving
24 + -11x + 4x2 = 10 + 5x

Solving for variable 'x'.

Reorder the terms:
24 + -10 + -11x + -5x + 4x2 = 10 + 5x + -10 + -5x

Combine like terms: 24 + -10 = 14
14 + -11x + -5x + 4x2 = 10 + 5x + -10 + -5x

Combine like terms: -11x + -5x = -16x
14 + -16x + 4x2 = 10 + 5x + -10 + -5x

Reorder the terms:
14 + -16x + 4x2 = 10 + -10 + 5x + -5x

Combine like terms: 10 + -10 = 0
14 + -16x + 4x2 = 0 + 5x + -5x
14 + -16x + 4x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
14 + -16x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(7 + -8x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(7 + -8x + 2x2)' equal to zero and attempt to solve:

Simplifying
7 + -8x + 2x2 = 0

Solving
7 + -8x + 2x2 = 0

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

Divide each side by '2'.
3.5 + -4x + x2 = 0

Move the constant term to the right:

Add '-3.5' to each side of the equation.
3.5 + -4x + -3.5 + x2 = 0 + -3.5

Reorder the terms:
3.5 + -3.5 + -4x + x2 = 0 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + -4x + x2 = 0 + -3.5
-4x + x2 = 0 + -3.5

Combine like terms: 0 + -3.5 = -3.5
-4x + x2 = -3.5

The x term is -4x.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4x + 4 + x2 = -3.5 + 4

Reorder the terms:
4 + -4x + x2 = -3.5 + 4

Combine like terms: -3.5 + 4 = 0.5
4 + -4x + x2 = 0.5

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

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(x + -2) equal to 0.707106781 and -0.707106781.


Subproblem 1
x + -2 = 0.707106781

Simplifying
x + -2 = 0.707106781

Reorder the terms:
-2 + x = 0.707106781

Solving
-2 + x = 0.707106781

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2' to each side of the equation.
-2 + 2 + x = 0.707106781 + 2

Combine like terms: -2 + 2 = 0
0 + x = 0.707106781 + 2
x = 0.707106781 + 2

Combine like terms: 0.707106781 + 2 = 2.707106781
x = 2.707106781

Simplifying
x = 2.707106781


Subproblem 2
x + -2 = -0.707106781

Simplifying
x + -2 = -0.707106781

Reorder the terms:
-2 + x = -0.707106781

Solving
-2 + x = -0.707106781

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2' to each side of the equation.
-2 + 2 + x = -0.707106781 + 2

Combine like terms: -2 + 2 = 0
0 + x = -0.707106781 + 2
x = -0.707106781 + 2

Combine like terms: -0.707106781 + 2 = 1.292893219
x = 1.292893219

Simplifying
x = 1.292893219


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.707106781, 1.292893219}
Solution
x = {2.707106781, 1.292893219}

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