a+7=a^2-25

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Solution for a+7=a^2-25 equation:


Simplifying
a + 7 = a2 + -25

Reorder the terms:
7 + a = a2 + -25

Reorder the terms:
7 + a = -25 + a2

Solving
7 + a = -25 + a2

Solving for variable 'a'.

Reorder the terms:
7 + 25 + a + -1a2 = -25 + a2 + 25 + -1a2

Combine like terms: 7 + 25 = 32
32 + a + -1a2 = -25 + a2 + 25 + -1a2

Reorder the terms:
32 + a + -1a2 = -25 + 25 + a2 + -1a2

Combine like terms: -25 + 25 = 0
32 + a + -1a2 = 0 + a2 + -1a2
32 + a + -1a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
32 + a + -1a2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-32 + -1a + a2 = 0

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + -1a + 32 + a2 = 0 + 32

Reorder the terms:
-32 + 32 + -1a + a2 = 0 + 32

Combine like terms: -32 + 32 = 0
0 + -1a + a2 = 0 + 32
-1a + a2 = 0 + 32

Combine like terms: 0 + 32 = 32
-1a + a2 = 32

The a term is a.  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.
-1a + 0.25 + a2 = 32 + 0.25

Reorder the terms:
0.25 + -1a + a2 = 32 + 0.25

Combine like terms: 32 + 0.25 = 32.25
0.25 + -1a + a2 = 32.25

Factor a perfect square on the left side:
(a + 0.5)(a + 0.5) = 32.25

Calculate the square root of the right side: 5.678908346

Break this problem into two subproblems by setting 
(a + 0.5) equal to 5.678908346 and -5.678908346.

Subproblem 1

a + 0.5 = 5.678908346 Simplifying a + 0.5 = 5.678908346 Reorder the terms: 0.5 + a = 5.678908346 Solving 0.5 + a = 5.678908346 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 5.678908346 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 5.678908346 + -0.5 a = 5.678908346 + -0.5 Combine like terms: 5.678908346 + -0.5 = 5.178908346 a = 5.178908346 Simplifying a = 5.178908346

Subproblem 2

a + 0.5 = -5.678908346 Simplifying a + 0.5 = -5.678908346 Reorder the terms: 0.5 + a = -5.678908346 Solving 0.5 + a = -5.678908346 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -5.678908346 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -5.678908346 + -0.5 a = -5.678908346 + -0.5 Combine like terms: -5.678908346 + -0.5 = -6.178908346 a = -6.178908346 Simplifying a = -6.178908346

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.178908346, -6.178908346}

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