(4a+1)(a+1)=(6-a)(3a-1)

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Solution for (4a+1)(a+1)=(6-a)(3a-1) equation:


Simplifying
(4a + 1)(a + 1) = (6 + -1a)(3a + -1)

Reorder the terms:
(1 + 4a)(a + 1) = (6 + -1a)(3a + -1)

Reorder the terms:
(1 + 4a)(1 + a) = (6 + -1a)(3a + -1)

Multiply (1 + 4a) * (1 + a)
(1(1 + a) + 4a * (1 + a)) = (6 + -1a)(3a + -1)
((1 * 1 + a * 1) + 4a * (1 + a)) = (6 + -1a)(3a + -1)
((1 + 1a) + 4a * (1 + a)) = (6 + -1a)(3a + -1)
(1 + 1a + (1 * 4a + a * 4a)) = (6 + -1a)(3a + -1)
(1 + 1a + (4a + 4a2)) = (6 + -1a)(3a + -1)

Combine like terms: 1a + 4a = 5a
(1 + 5a + 4a2) = (6 + -1a)(3a + -1)

Reorder the terms:
1 + 5a + 4a2 = (6 + -1a)(-1 + 3a)

Multiply (6 + -1a) * (-1 + 3a)
1 + 5a + 4a2 = (6(-1 + 3a) + -1a * (-1 + 3a))
1 + 5a + 4a2 = ((-1 * 6 + 3a * 6) + -1a * (-1 + 3a))
1 + 5a + 4a2 = ((-6 + 18a) + -1a * (-1 + 3a))
1 + 5a + 4a2 = (-6 + 18a + (-1 * -1a + 3a * -1a))
1 + 5a + 4a2 = (-6 + 18a + (1a + -3a2))

Combine like terms: 18a + 1a = 19a
1 + 5a + 4a2 = (-6 + 19a + -3a2)

Solving
1 + 5a + 4a2 = -6 + 19a + -3a2

Solving for variable 'a'.

Reorder the terms:
1 + 6 + 5a + -19a + 4a2 + 3a2 = -6 + 19a + -3a2 + 6 + -19a + 3a2

Combine like terms: 1 + 6 = 7
7 + 5a + -19a + 4a2 + 3a2 = -6 + 19a + -3a2 + 6 + -19a + 3a2

Combine like terms: 5a + -19a = -14a
7 + -14a + 4a2 + 3a2 = -6 + 19a + -3a2 + 6 + -19a + 3a2

Combine like terms: 4a2 + 3a2 = 7a2
7 + -14a + 7a2 = -6 + 19a + -3a2 + 6 + -19a + 3a2

Reorder the terms:
7 + -14a + 7a2 = -6 + 6 + 19a + -19a + -3a2 + 3a2

Combine like terms: -6 + 6 = 0
7 + -14a + 7a2 = 0 + 19a + -19a + -3a2 + 3a2
7 + -14a + 7a2 = 19a + -19a + -3a2 + 3a2

Combine like terms: 19a + -19a = 0
7 + -14a + 7a2 = 0 + -3a2 + 3a2
7 + -14a + 7a2 = -3a2 + 3a2

Combine like terms: -3a2 + 3a2 = 0
7 + -14a + 7a2 = 0

Factor out the Greatest Common Factor (GCF), '7'.
7(1 + -2a + a2) = 0

Factor a trinomial.
7((1 + -1a)(1 + -1a)) = 0

Ignore the factor 7.

Subproblem 1

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

Subproblem 2

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

Solution

a = {1, 1}

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