(a)(a)+(a+1)(a+1)+(a+2)(a+2)=(a+3)(a+3)+(a+4)(a+4)

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Solution for (a)(a)+(a+1)(a+1)+(a+2)(a+2)=(a+3)(a+3)+(a+4)(a+4) equation:


Simplifying
(a)(a) + (a + 1)(a + 1) + (a + 2)(a + 2) = (a + 3)(a + 3) + (a + 4)(a + 4)

Multiply a * a
a2 + (a + 1)(a + 1) + (a + 2)(a + 2) = (a + 3)(a + 3) + (a + 4)(a + 4)

Reorder the terms:
a2 + (1 + a)(a + 1) + (a + 2)(a + 2) = (a + 3)(a + 3) + (a + 4)(a + 4)

Reorder the terms:
a2 + (1 + a)(1 + a) + (a + 2)(a + 2) = (a + 3)(a + 3) + (a + 4)(a + 4)

Multiply (1 + a) * (1 + a)
a2 + (1(1 + a) + a(1 + a)) + (a + 2)(a + 2) = (a + 3)(a + 3) + (a + 4)(a + 4)
a2 + ((1 * 1 + a * 1) + a(1 + a)) + (a + 2)(a + 2) = (a + 3)(a + 3) + (a + 4)(a + 4)
a2 + ((1 + 1a) + a(1 + a)) + (a + 2)(a + 2) = (a + 3)(a + 3) + (a + 4)(a + 4)
a2 + (1 + 1a + (1 * a + a * a)) + (a + 2)(a + 2) = (a + 3)(a + 3) + (a + 4)(a + 4)
a2 + (1 + 1a + (1a + a2)) + (a + 2)(a + 2) = (a + 3)(a + 3) + (a + 4)(a + 4)

Combine like terms: 1a + 1a = 2a
a2 + (1 + 2a + a2) + (a + 2)(a + 2) = (a + 3)(a + 3) + (a + 4)(a + 4)

Reorder the terms:
a2 + 1 + 2a + a2 + (2 + a)(a + 2) = (a + 3)(a + 3) + (a + 4)(a + 4)

Reorder the terms:
a2 + 1 + 2a + a2 + (2 + a)(2 + a) = (a + 3)(a + 3) + (a + 4)(a + 4)

Multiply (2 + a) * (2 + a)
a2 + 1 + 2a + a2 + (2(2 + a) + a(2 + a)) = (a + 3)(a + 3) + (a + 4)(a + 4)
a2 + 1 + 2a + a2 + ((2 * 2 + a * 2) + a(2 + a)) = (a + 3)(a + 3) + (a + 4)(a + 4)
a2 + 1 + 2a + a2 + ((4 + 2a) + a(2 + a)) = (a + 3)(a + 3) + (a + 4)(a + 4)
a2 + 1 + 2a + a2 + (4 + 2a + (2 * a + a * a)) = (a + 3)(a + 3) + (a + 4)(a + 4)
a2 + 1 + 2a + a2 + (4 + 2a + (2a + a2)) = (a + 3)(a + 3) + (a + 4)(a + 4)

Combine like terms: 2a + 2a = 4a
a2 + 1 + 2a + a2 + (4 + 4a + a2) = (a + 3)(a + 3) + (a + 4)(a + 4)

Reorder the terms:
1 + 4 + 2a + 4a + a2 + a2 + a2 = (a + 3)(a + 3) + (a + 4)(a + 4)

Combine like terms: 1 + 4 = 5
5 + 2a + 4a + a2 + a2 + a2 = (a + 3)(a + 3) + (a + 4)(a + 4)

Combine like terms: 2a + 4a = 6a
5 + 6a + a2 + a2 + a2 = (a + 3)(a + 3) + (a + 4)(a + 4)

Combine like terms: a2 + a2 = 2a2
5 + 6a + 2a2 + a2 = (a + 3)(a + 3) + (a + 4)(a + 4)

Combine like terms: 2a2 + a2 = 3a2
5 + 6a + 3a2 = (a + 3)(a + 3) + (a + 4)(a + 4)

Reorder the terms:
5 + 6a + 3a2 = (3 + a)(a + 3) + (a + 4)(a + 4)

Reorder the terms:
5 + 6a + 3a2 = (3 + a)(3 + a) + (a + 4)(a + 4)

Multiply (3 + a) * (3 + a)
5 + 6a + 3a2 = (3(3 + a) + a(3 + a)) + (a + 4)(a + 4)
5 + 6a + 3a2 = ((3 * 3 + a * 3) + a(3 + a)) + (a + 4)(a + 4)
5 + 6a + 3a2 = ((9 + 3a) + a(3 + a)) + (a + 4)(a + 4)
5 + 6a + 3a2 = (9 + 3a + (3 * a + a * a)) + (a + 4)(a + 4)
5 + 6a + 3a2 = (9 + 3a + (3a + a2)) + (a + 4)(a + 4)

Combine like terms: 3a + 3a = 6a
5 + 6a + 3a2 = (9 + 6a + a2) + (a + 4)(a + 4)

Reorder the terms:
5 + 6a + 3a2 = 9 + 6a + a2 + (4 + a)(a + 4)

Reorder the terms:
5 + 6a + 3a2 = 9 + 6a + a2 + (4 + a)(4 + a)

Multiply (4 + a) * (4 + a)
5 + 6a + 3a2 = 9 + 6a + a2 + (4(4 + a) + a(4 + a))
5 + 6a + 3a2 = 9 + 6a + a2 + ((4 * 4 + a * 4) + a(4 + a))
5 + 6a + 3a2 = 9 + 6a + a2 + ((16 + 4a) + a(4 + a))
5 + 6a + 3a2 = 9 + 6a + a2 + (16 + 4a + (4 * a + a * a))
5 + 6a + 3a2 = 9 + 6a + a2 + (16 + 4a + (4a + a2))

Combine like terms: 4a + 4a = 8a
5 + 6a + 3a2 = 9 + 6a + a2 + (16 + 8a + a2)

Reorder the terms:
5 + 6a + 3a2 = 9 + 16 + 6a + 8a + a2 + a2

Combine like terms: 9 + 16 = 25
5 + 6a + 3a2 = 25 + 6a + 8a + a2 + a2

Combine like terms: 6a + 8a = 14a
5 + 6a + 3a2 = 25 + 14a + a2 + a2

Combine like terms: a2 + a2 = 2a2
5 + 6a + 3a2 = 25 + 14a + 2a2

Solving
5 + 6a + 3a2 = 25 + 14a + 2a2

Solving for variable 'a'.

Reorder the terms:
5 + -25 + 6a + -14a + 3a2 + -2a2 = 25 + 14a + 2a2 + -25 + -14a + -2a2

Combine like terms: 5 + -25 = -20
-20 + 6a + -14a + 3a2 + -2a2 = 25 + 14a + 2a2 + -25 + -14a + -2a2

Combine like terms: 6a + -14a = -8a
-20 + -8a + 3a2 + -2a2 = 25 + 14a + 2a2 + -25 + -14a + -2a2

Combine like terms: 3a2 + -2a2 = 1a2
-20 + -8a + 1a2 = 25 + 14a + 2a2 + -25 + -14a + -2a2

Reorder the terms:
-20 + -8a + 1a2 = 25 + -25 + 14a + -14a + 2a2 + -2a2

Combine like terms: 25 + -25 = 0
-20 + -8a + 1a2 = 0 + 14a + -14a + 2a2 + -2a2
-20 + -8a + 1a2 = 14a + -14a + 2a2 + -2a2

Combine like terms: 14a + -14a = 0
-20 + -8a + 1a2 = 0 + 2a2 + -2a2
-20 + -8a + 1a2 = 2a2 + -2a2

Combine like terms: 2a2 + -2a2 = 0
-20 + -8a + 1a2 = 0

Factor a trinomial.
(-2 + -1a)(10 + -1a) = 0

Subproblem 1

Set the factor '(-2 + -1a)' equal to zero and attempt to solve: Simplifying -2 + -1a = 0 Solving -2 + -1a = 0 Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + -1a = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -1a = 0 + 2 -1a = 0 + 2 Combine like terms: 0 + 2 = 2 -1a = 2 Divide each side by '-1'. a = -2 Simplifying a = -2

Subproblem 2

Set the factor '(10 + -1a)' equal to zero and attempt to solve: Simplifying 10 + -1a = 0 Solving 10 + -1a = 0 Move all terms containing a to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + -1a = 0 + -10 Combine like terms: 10 + -10 = 0 0 + -1a = 0 + -10 -1a = 0 + -10 Combine like terms: 0 + -10 = -10 -1a = -10 Divide each side by '-1'. a = 10 Simplifying a = 10

Solution

a = {-2, 10}

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