a^2+2a-23=0

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


Simplifying
a2 + 2a + -23 = 0

Reorder the terms:
-23 + 2a + a2 = 0

Solving
-23 + 2a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + 2a + 23 + a2 = 0 + 23

Reorder the terms:
-23 + 23 + 2a + a2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + 2a + a2 = 0 + 23
2a + a2 = 0 + 23

Combine like terms: 0 + 23 = 23
2a + a2 = 23

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

Add '1' to each side of the equation.
2a + 1 + a2 = 23 + 1

Reorder the terms:
1 + 2a + a2 = 23 + 1

Combine like terms: 23 + 1 = 24
1 + 2a + a2 = 24

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

Calculate the square root of the right side: 4.898979486

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

Subproblem 1

a + 1 = 4.898979486 Simplifying a + 1 = 4.898979486 Reorder the terms: 1 + a = 4.898979486 Solving 1 + a = 4.898979486 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = 4.898979486 + -1 Combine like terms: 1 + -1 = 0 0 + a = 4.898979486 + -1 a = 4.898979486 + -1 Combine like terms: 4.898979486 + -1 = 3.898979486 a = 3.898979486 Simplifying a = 3.898979486

Subproblem 2

a + 1 = -4.898979486 Simplifying a + 1 = -4.898979486 Reorder the terms: 1 + a = -4.898979486 Solving 1 + a = -4.898979486 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = -4.898979486 + -1 Combine like terms: 1 + -1 = 0 0 + a = -4.898979486 + -1 a = -4.898979486 + -1 Combine like terms: -4.898979486 + -1 = -5.898979486 a = -5.898979486 Simplifying a = -5.898979486

Solution

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

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