0=-x^2+2x+23

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Solution for 0=-x^2+2x+23 equation:


Simplifying
0 = -1x2 + 2x + 23

Reorder the terms:
0 = 23 + 2x + -1x2

Solving
0 = 23 + 2x + -1x2

Solving for variable 'x'.

Combine like terms: 0 + -23 = -23
-23 + -2x + x2 = 23 + 2x + -1x2 + -23 + -2x + x2

Reorder the terms:
-23 + -2x + x2 = 23 + -23 + 2x + -2x + -1x2 + x2

Combine like terms: 23 + -23 = 0
-23 + -2x + x2 = 0 + 2x + -2x + -1x2 + x2
-23 + -2x + x2 = 2x + -2x + -1x2 + x2

Combine like terms: 2x + -2x = 0
-23 + -2x + x2 = 0 + -1x2 + x2
-23 + -2x + x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
-23 + -2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + -2x + 23 + x2 = 0 + 23

Reorder the terms:
-23 + 23 + -2x + x2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + -2x + x2 = 0 + 23
-2x + x2 = 0 + 23

Combine like terms: 0 + 23 = 23
-2x + x2 = 23

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 = 23 + 1

Reorder the terms:
1 + -2x + x2 = 23 + 1

Combine like terms: 23 + 1 = 24
1 + -2x + x2 = 24

Factor a perfect square on the left side:
(x + -1)(x + -1) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(x + -1) equal to 4.898979486 and -4.898979486.

Subproblem 1

x + -1 = 4.898979486 Simplifying x + -1 = 4.898979486 Reorder the terms: -1 + x = 4.898979486 Solving -1 + x = 4.898979486 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 = 4.898979486 + 1 Combine like terms: -1 + 1 = 0 0 + x = 4.898979486 + 1 x = 4.898979486 + 1 Combine like terms: 4.898979486 + 1 = 5.898979486 x = 5.898979486 Simplifying x = 5.898979486

Subproblem 2

x + -1 = -4.898979486 Simplifying x + -1 = -4.898979486 Reorder the terms: -1 + x = -4.898979486 Solving -1 + x = -4.898979486 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 = -4.898979486 + 1 Combine like terms: -1 + 1 = 0 0 + x = -4.898979486 + 1 x = -4.898979486 + 1 Combine like terms: -4.898979486 + 1 = -3.898979486 x = -3.898979486 Simplifying x = -3.898979486

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.898979486, -3.898979486}

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