'a' and 'b'are integers and (a, b) = 2. Solve the following two equations:

(a+b)^2 = 25a
18(a^4) + (b^4) = 41a(b^2)

Can't get it, help please?

To solve the given equations, we will use the fact that (a, b) = 2.

First, let's simplify the equations step by step:

1. (a+b)^2 = 25a

Expanding the left side of the equation, we get:

a^2 + 2ab + b^2 = 25a

Rearranging this equation, we get:

a^2 - 25a + 2ab + b^2 = 0 ----- (Equation 1)

2. 18(a^4) + (b^4) = 41a(b^2)

As both sides of the equation have a common factor of a, we can divide both sides by a to simplify:

18a^3 + b^4 = 41b^2

Rearranging this equation, we get:

b^4 - 41b^2 + 18a^3 = 0 ----- (Equation 2)

Now, we have two equations (Equation 1 and Equation 2) in two variables (a and b). To solve these equations simultaneously, we will use a method called substitution.

Let's solve Equation 1 for 'b':

b^2 + 2ab + (a^2 - 25a) = 0

Rearranging this equation, we get:

b^2 + 2ab + a^2 - 25a = 0

Using the quadratic formula, we have:

b = (-2a ± √((2a)^2 - 4(a^2 - 25a)))/(2)

b = (-2a ± √(4a^2 - 4a^2 + 100a))/(2)

b = (-2a ± √(100a))/(2)

b = (-a ± √(100a))/2

We know that (a, b) = 2, so we can substitute this value into the equation:

2 = (-a ± √(100a))/2

Multiplying both sides by 2, we get:

4 = -a ± √(100a)

Now, we have two cases to consider:

Case 1: When (-a + √(100a)) = 4

Squaring both sides of the equation, we get:

(-a + √(100a))^2 = 16

a^2 - 2a√(100a) + 100a = 16

Rearranging this equation, we get:

101a - 2a√(100a) = 16

Dividing both sides by a, we get:

101 - 2√(100a) = 16/a

2√(100a) = 101 - 16/a

Squaring both sides of the equation, we get:

4(100a) = 101^2 - 32(101) + (16/a^2)

400a = 101^2 - 32(101) + (16/a^2)

Substituting a = 2, we get:

800 = 101^2 - 32(101) + (16/4)

800 = 101^2 - 3232 + 4

800 = 10201 - 3232 + 4

800 = 6973

The equation is not satisfied in this case.

Case 2: When (-a - √(100a)) = 4

Squaring both sides of the equation, we get:

(-a - √(100a))^2 = 16

a^2 + 2a√(100a) + 100a = 16

Rearranging this equation, we get:

101a + 2a√(100a) = 16

Dividing both sides by a, we get:

101 + 2√(100a) = 16/a

2√(100a) = 16/a - 101

Squaring both sides of the equation, we get:

4(100a) = (16/a - 101)^2

400a = (16^2/a^2) + 101^2 - 2(16)(101/a)

400a = 256/a^2 + 10201 - 3232/a

Multiplying both sides by a^2, we get:

400a^3 = 256 + 10201a^2 - 3232a

Rearranging the equation, we get:

400a^3 - 10201a^2 + 3232a - 256 = 0

Now, we have a cubic equation in terms of 'a'. We can solve this equation numerically using computer software or calculators that have the capability to solve cubic equations.

Once we find the value(s) of 'a', we can substitute it back into Equation 1 or Equation 2 to solve for 'b'.

Please note that since we did not obtain a specific value for 'a' in this explanation, the final step of finding 'b' cannot be performed until the value of 'a' is known. In such cases, numerical methods as mentioned above are generally needed to obtain approximate solutions.

I hope this explanation helps you understand how to approach solving the given set of equations.