45-3x^2+x=24

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Solution for 45-3x^2+x=24 equation:


Simplifying
45 + -3x2 + x = 24

Reorder the terms:
45 + x + -3x2 = 24

Solving
45 + x + -3x2 = 24

Solving for variable 'x'.

Reorder the terms:
45 + -24 + x + -3x2 = 24 + -24

Combine like terms: 45 + -24 = 21
21 + x + -3x2 = 24 + -24

Combine like terms: 24 + -24 = 0
21 + x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-7 + -0.3333333333x + x2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -0.3333333333x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + -0.3333333333x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -0.3333333333x + x2 = 0 + 7
-0.3333333333x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
-0.3333333333x + x2 = 7

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-0.3333333333x + 0.25 + x2 = 7 + 0.25

Reorder the terms:
0.25 + -0.3333333333x + x2 = 7 + 0.25

Combine like terms: 7 + 0.25 = 7.25
0.25 + -0.3333333333x + x2 = 7.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

x + 0.5 = 2.692582404 Simplifying x + 0.5 = 2.692582404 Reorder the terms: 0.5 + x = 2.692582404 Solving 0.5 + x = 2.692582404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 2.692582404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.692582404 + -0.5 x = 2.692582404 + -0.5 Combine like terms: 2.692582404 + -0.5 = 2.192582404 x = 2.192582404 Simplifying x = 2.192582404

Subproblem 2

x + 0.5 = -2.692582404 Simplifying x + 0.5 = -2.692582404 Reorder the terms: 0.5 + x = -2.692582404 Solving 0.5 + x = -2.692582404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -2.692582404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.692582404 + -0.5 x = -2.692582404 + -0.5 Combine like terms: -2.692582404 + -0.5 = -3.192582404 x = -3.192582404 Simplifying x = -3.192582404

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.192582404, -3.192582404}

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