2x^2=24x+45

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Solution for 2x^2=24x+45 equation:


Simplifying
2x2 = 24x + 45

Reorder the terms:
2x2 = 45 + 24x

Solving
2x2 = 45 + 24x

Solving for variable 'x'.

Reorder the terms:
-45 + -24x + 2x2 = 45 + 24x + -45 + -24x

Reorder the terms:
-45 + -24x + 2x2 = 45 + -45 + 24x + -24x

Combine like terms: 45 + -45 = 0
-45 + -24x + 2x2 = 0 + 24x + -24x
-45 + -24x + 2x2 = 24x + -24x

Combine like terms: 24x + -24x = 0
-45 + -24x + 2x2 = 0

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

Divide each side by '2'.
-22.5 + -12x + x2 = 0

Move the constant term to the right:

Add '22.5' to each side of the equation.
-22.5 + -12x + 22.5 + x2 = 0 + 22.5

Reorder the terms:
-22.5 + 22.5 + -12x + x2 = 0 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + -12x + x2 = 0 + 22.5
-12x + x2 = 0 + 22.5

Combine like terms: 0 + 22.5 = 22.5
-12x + x2 = 22.5

The x term is -12x.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12x + 36 + x2 = 22.5 + 36

Reorder the terms:
36 + -12x + x2 = 22.5 + 36

Combine like terms: 22.5 + 36 = 58.5
36 + -12x + x2 = 58.5

Factor a perfect square on the left side:
(x + -6)(x + -6) = 58.5

Calculate the square root of the right side: 7.64852927

Break this problem into two subproblems by setting 
(x + -6) equal to 7.64852927 and -7.64852927.

Subproblem 1

x + -6 = 7.64852927 Simplifying x + -6 = 7.64852927 Reorder the terms: -6 + x = 7.64852927 Solving -6 + x = 7.64852927 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + x = 7.64852927 + 6 Combine like terms: -6 + 6 = 0 0 + x = 7.64852927 + 6 x = 7.64852927 + 6 Combine like terms: 7.64852927 + 6 = 13.64852927 x = 13.64852927 Simplifying x = 13.64852927

Subproblem 2

x + -6 = -7.64852927 Simplifying x + -6 = -7.64852927 Reorder the terms: -6 + x = -7.64852927 Solving -6 + x = -7.64852927 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + x = -7.64852927 + 6 Combine like terms: -6 + 6 = 0 0 + x = -7.64852927 + 6 x = -7.64852927 + 6 Combine like terms: -7.64852927 + 6 = -1.64852927 x = -1.64852927 Simplifying x = -1.64852927

Solution

The solution to the problem is based on the solutions from the subproblems. x = {13.64852927, -1.64852927}

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