24x^2=55x+89

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Solution for 24x^2=55x+89 equation:


Simplifying
24x2 = 55x + 89

Reorder the terms:
24x2 = 89 + 55x

Solving
24x2 = 89 + 55x

Solving for variable 'x'.

Reorder the terms:
-89 + -55x + 24x2 = 89 + 55x + -89 + -55x

Reorder the terms:
-89 + -55x + 24x2 = 89 + -89 + 55x + -55x

Combine like terms: 89 + -89 = 0
-89 + -55x + 24x2 = 0 + 55x + -55x
-89 + -55x + 24x2 = 55x + -55x

Combine like terms: 55x + -55x = 0
-89 + -55x + 24x2 = 0

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

Divide each side by '24'.
-3.708333333 + -2.291666667x + x2 = 0

Move the constant term to the right:

Add '3.708333333' to each side of the equation.
-3.708333333 + -2.291666667x + 3.708333333 + x2 = 0 + 3.708333333

Reorder the terms:
-3.708333333 + 3.708333333 + -2.291666667x + x2 = 0 + 3.708333333

Combine like terms: -3.708333333 + 3.708333333 = 0.000000000
0.000000000 + -2.291666667x + x2 = 0 + 3.708333333
-2.291666667x + x2 = 0 + 3.708333333

Combine like terms: 0 + 3.708333333 = 3.708333333
-2.291666667x + x2 = 3.708333333

The x term is -2.291666667x.  Take half its coefficient (-1.145833334).
Square it (1.312934029) and add it to both sides.

Add '1.312934029' to each side of the equation.
-2.291666667x + 1.312934029 + x2 = 3.708333333 + 1.312934029

Reorder the terms:
1.312934029 + -2.291666667x + x2 = 3.708333333 + 1.312934029

Combine like terms: 3.708333333 + 1.312934029 = 5.021267362
1.312934029 + -2.291666667x + x2 = 5.021267362

Factor a perfect square on the left side:
(x + -1.145833334)(x + -1.145833334) = 5.021267362

Calculate the square root of the right side: 2.240818458

Break this problem into two subproblems by setting 
(x + -1.145833334) equal to 2.240818458 and -2.240818458.

Subproblem 1

x + -1.145833334 = 2.240818458 Simplifying x + -1.145833334 = 2.240818458 Reorder the terms: -1.145833334 + x = 2.240818458 Solving -1.145833334 + x = 2.240818458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.145833334' to each side of the equation. -1.145833334 + 1.145833334 + x = 2.240818458 + 1.145833334 Combine like terms: -1.145833334 + 1.145833334 = 0.000000000 0.000000000 + x = 2.240818458 + 1.145833334 x = 2.240818458 + 1.145833334 Combine like terms: 2.240818458 + 1.145833334 = 3.386651792 x = 3.386651792 Simplifying x = 3.386651792

Subproblem 2

x + -1.145833334 = -2.240818458 Simplifying x + -1.145833334 = -2.240818458 Reorder the terms: -1.145833334 + x = -2.240818458 Solving -1.145833334 + x = -2.240818458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.145833334' to each side of the equation. -1.145833334 + 1.145833334 + x = -2.240818458 + 1.145833334 Combine like terms: -1.145833334 + 1.145833334 = 0.000000000 0.000000000 + x = -2.240818458 + 1.145833334 x = -2.240818458 + 1.145833334 Combine like terms: -2.240818458 + 1.145833334 = -1.094985124 x = -1.094985124 Simplifying x = -1.094985124

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.386651792, -1.094985124}

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