Algebra Problem with Rationals and Proofs

• silvermane
In summary, the conversation discusses two proofs, one regarding the equality of two expressions with a radical and the other regarding the non-zero nature of an expression with rational numbers. The conversation provides some tips and guidance for approaching the proofs, ultimately leading to a solution.
silvermane
Gold Member
Hello fellow forum buddies :)

Homework Statement

a.) Prove that if a+b$$\sqrt{2}$$ = c+d$$\sqrt{2}$$ with a,b,c,d all in Q, then
a = c and b = d.
b.) Prove that a^2 - 2b^2 with a, b in Q is nonzero unless a=b=0

The Attempt at a Solution

I really don't know where to start. Any tips would be nice. I just need to see something that I'm not seeing at the moment. :(
I don't need an answer, just a little jolt lol. It will be greatly appreciated! :)

For part a, is (a-c) in Q? Is (b-d)$$\sqrt{2}$$?

For part b, factor the equation and try to use the result from part a.

I guess he ^ already took care of you ;-).

Post again if you need more clearification.

fzero said:
For part a, is (a-c) in Q? Is (b-d)$$\sqrt{2}$$?

For part b, factor the equation and try to use the result from part a.

AHHH... wow I'm glad I see it now. So I just need to solve a and c with respect to the fact that they are rational, then b and d with the fact that they're not rational?

╔(σ_σ)╝ said:
I guess he ^ already took care of you ;-).

Post again if you need more clearification.

Awe phanku :)

silvermane said:
AHHH... wow I'm glad I see it now. So I just need to solve a and c with respect to the fact that they are rational, then b and d with the fact that they're not rational?

b and d are rational, however, the product with sqrt(2) is not. I believe this is what you meant.

I believe you can see that if b-d was non-zero a contradiction ensues. :-)

EDIT

lmao.
U r velcome :-).

silvermane said:
AHHH... wow I'm glad I see it now. So I just need to solve a and c with respect to the fact that they are rational, then b and d with the fact that they're not rational?

Well if x = a - c is nonzero, can we find y such that x = y $$\sqrt{2}$$? Is such a y compatible with y = b-d?

Ahhhh that should do it lol. I wrote that down and when I saw each of your replies, it was confirmed. I think I can finish it from here. Thanks so much! :)

Okay. Post if you get stuck in part b or something.

1. What are rational numbers?

Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not equal to zero. They include both positive and negative fractions, as well as whole numbers and integers.

2. How do you solve algebra problems with rationals?

To solve algebra problems with rationals, you can use the same techniques as solving any other algebraic equation. This includes simplifying fractions, using the distributive property, and combining like terms. It is also important to follow the order of operations when solving these equations.

3. What are some common mistakes people make when solving algebra problems with rationals?

One common mistake is forgetting to simplify fractions before solving the equation. This can result in incorrect answers. Another mistake is not following the order of operations, which can also lead to incorrect solutions. It is important to pay attention to the rules of algebra and to double-check your work when solving these problems.

4. Can you use proofs when solving algebra problems with rationals?

Yes, proofs can be used to solve algebra problems with rationals. Proofs involve using logical reasoning and mathematical properties to show that an equation or statement is true. This can be particularly helpful when dealing with complex algebra problems involving rationals.

5. How can I check my answer for an algebra problem with rationals?

To check your answer for an algebra problem with rationals, you can substitute your solution back into the original equation and see if it satisfies the equation. You can also verify your solution by using a calculator to evaluate the equation with your solution plugged in. Another helpful method is to graph the equation and see if your solution lies on the graph.

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