a^2-410a+21=0

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Solution for a^2-410a+21=0 equation:


Simplifying
a2 + -410a + 21 = 0

Reorder the terms:
21 + -410a + a2 = 0

Solving
21 + -410a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '-21' to each side of the equation.
21 + -410a + -21 + a2 = 0 + -21

Reorder the terms:
21 + -21 + -410a + a2 = 0 + -21

Combine like terms: 21 + -21 = 0
0 + -410a + a2 = 0 + -21
-410a + a2 = 0 + -21

Combine like terms: 0 + -21 = -21
-410a + a2 = -21

The a term is -410a.  Take half its coefficient (-205).
Square it (42025) and add it to both sides.

Add '42025' to each side of the equation.
-410a + 42025 + a2 = -21 + 42025

Reorder the terms:
42025 + -410a + a2 = -21 + 42025

Combine like terms: -21 + 42025 = 42004
42025 + -410a + a2 = 42004

Factor a perfect square on the left side:
(a + -205)(a + -205) = 42004

Calculate the square root of the right side: 204.948774088

Break this problem into two subproblems by setting 
(a + -205) equal to 204.948774088 and -204.948774088.

Subproblem 1

a + -205 = 204.948774088 Simplifying a + -205 = 204.948774088 Reorder the terms: -205 + a = 204.948774088 Solving -205 + a = 204.948774088 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '205' to each side of the equation. -205 + 205 + a = 204.948774088 + 205 Combine like terms: -205 + 205 = 0 0 + a = 204.948774088 + 205 a = 204.948774088 + 205 Combine like terms: 204.948774088 + 205 = 409.948774088 a = 409.948774088 Simplifying a = 409.948774088

Subproblem 2

a + -205 = -204.948774088 Simplifying a + -205 = -204.948774088 Reorder the terms: -205 + a = -204.948774088 Solving -205 + a = -204.948774088 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '205' to each side of the equation. -205 + 205 + a = -204.948774088 + 205 Combine like terms: -205 + 205 = 0 0 + a = -204.948774088 + 205 a = -204.948774088 + 205 Combine like terms: -204.948774088 + 205 = 0.051225912 a = 0.051225912 Simplifying a = 0.051225912

Solution

The solution to the problem is based on the solutions from the subproblems. a = {409.948774088, 0.051225912}

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