23.22=-0.5x^2+x+24

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Solution for 23.22=-0.5x^2+x+24 equation:


Simplifying
23.22 = -0.5x2 + x + 24

Reorder the terms:
23.22 = 24 + x + -0.5x2

Solving
23.22 = 24 + x + -0.5x2

Solving for variable 'x'.

Combine like terms: 23.22 + -24 = -0.78
-0.78 + -1x + 0.5x2 = 24 + x + -0.5x2 + -24 + -1x + 0.5x2

Reorder the terms:
-0.78 + -1x + 0.5x2 = 24 + -24 + x + -1x + -0.5x2 + 0.5x2

Combine like terms: 24 + -24 = 0
-0.78 + -1x + 0.5x2 = 0 + x + -1x + -0.5x2 + 0.5x2
-0.78 + -1x + 0.5x2 = x + -1x + -0.5x2 + 0.5x2

Combine like terms: x + -1x = 0
-0.78 + -1x + 0.5x2 = 0 + -0.5x2 + 0.5x2
-0.78 + -1x + 0.5x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
-0.78 + -1x + 0.5x2 = 0.0

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

Divide each side by '0.5'.
-1.56 + -2x + x2 = 0

Move the constant term to the right:

Add '1.56' to each side of the equation.
-1.56 + -2x + 1.56 + x2 = 0 + 1.56

Reorder the terms:
-1.56 + 1.56 + -2x + x2 = 0 + 1.56

Combine like terms: -1.56 + 1.56 = 0.00
0.00 + -2x + x2 = 0 + 1.56
-2x + x2 = 0 + 1.56

Combine like terms: 0 + 1.56 = 1.56
-2x + x2 = 1.56

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 1.56 + 1

Reorder the terms:
1 + -2x + x2 = 1.56 + 1

Combine like terms: 1.56 + 1 = 2.56
1 + -2x + x2 = 2.56

Factor a perfect square on the left side:
(x + -1)(x + -1) = 2.56

Calculate the square root of the right side: 1.6

Break this problem into two subproblems by setting 
(x + -1) equal to 1.6 and -1.6.

Subproblem 1

x + -1 = 1.6 Simplifying x + -1 = 1.6 Reorder the terms: -1 + x = 1.6 Solving -1 + x = 1.6 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 1.6 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.6 + 1 x = 1.6 + 1 Combine like terms: 1.6 + 1 = 2.6 x = 2.6 Simplifying x = 2.6

Subproblem 2

x + -1 = -1.6 Simplifying x + -1 = -1.6 Reorder the terms: -1 + x = -1.6 Solving -1 + x = -1.6 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -1.6 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.6 + 1 x = -1.6 + 1 Combine like terms: -1.6 + 1 = -0.6 x = -0.6 Simplifying x = -0.6

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.6, -0.6}

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