How to Solve Complex Algebraic Equations Involving Square Roots?

[abrk]

Hello.
I've been trying to crack these three for half a day and decided to ask for help.

1) \(\displaystyle (a+b(\sqrt{2})^2=33+20(\sqrt{2})\)

2) \(\displaystyle (a+b(\sqrt{2})^2=41-24(\sqrt{2})\)

3) Find: \(\displaystyle \sqrt{11-6\sqrt{2}}\).

 

[HallsofIvy]

Hello.
I've been trying to crack these three for half a day and decided to ask for help.

1) \(\displaystyle (a+b(\sqrt{2})^2=33+20(\sqrt{2})\)

2) \(\displaystyle (a+b(\sqrt{2})^2=41-24(\sqrt{2})\)

3) Find: \(\displaystyle \sqrt{11-6\sqrt{2}}\).

You are missing a ")" in the first two problems. Since just \(\displaystyle (\sqrt{2})^2\) would make the problem trivial I assume you mean \(\displaystyle (a+ b\sqrt{2})^2\) in each.

Okay, in those two just go ahead and square on the left. I presume you know that \(\displaystyle (x+ y)^2= x^2+ 2xy+ y^2\).
\(\displaystyle (a+ b \sqrt{2})^2= a^2+ 2ab\sqrt{2}+ 2b^2\)
So in (1) you have \(\displaystyle a^2+ 2b^2+ 2ab\sqrt{2}= 33+ 20\sqrt{2}\). Assuming that a and b are supposed to be rational numbers, you must have \(\displaystyle a^2+ 2b= 33\) and \(\displaystyle 2ab= 20\)

For (3), I don't know what you mean by "find" that number. The way it is given is perfectly valid. I presume you do NOT mean just use a calculator!

 

[Petrus]

\(\displaystyle a^2+ 2b= 33\)

Hello,
You forgot \(\displaystyle a^2+2b^2=33\), just wanted to tell that you forgot that so you can Edit! Have a nice day!
Regards,
\(\displaystyle |\pi\rangle\)

 

[abrk]

You are missing a ")" in the first two problems. Since just \(\displaystyle (\sqrt{2})^2\) would make the problem trivial I assume you mean \(\displaystyle (a+ b\sqrt{2})^2\) in each.

Okay, in those two just go ahead and square on the left. I presume you know that \(\displaystyle (x+ y)^2= x^2+ 2xy+ y^2\).
\(\displaystyle (a+ b \sqrt{2})^2= a^2+ 2ab\sqrt{2}+ 2b^2\)
So in (1) you have \(\displaystyle a^2+ 2b^2+ 2ab\sqrt{2}= 33+ 20\sqrt{2}\). Assuming that a and b are supposed to be rational numbers, you must have \(\displaystyle a^2+ 2b= 33\) and \(\displaystyle 2ab= 20\)

For (3), I don't know what you mean by "find" that number. The way it is given is perfectly valid. I presume you do NOT mean just use a calculator!

For 1 and 2. Indeed, there was one more bracket, which I didn't embed in my original question. My whole problem is that I cannot get through the equation system

\(\displaystyle a^2+b^2=c\)
\(\displaystyle a*b=d\)

Or in this case \(\displaystyle a^2+2b^2=33\) and \(\displaystyle a*b=20\)

In 3, that's the whole description of the task, but a assume that it is to be represented as \(\displaystyle a+b\sqrt{2}\), thus it is exactly the same as the first two problems.

 

[Petrus]

Hello,
we got that:
(1)\(\displaystyle a^2+2b^2=33\)
(2)\(\displaystyle a*b=20\)
from (2) we get that \(\displaystyle a=\frac{20}{b}\) and put that into (1) and solve it!
Good luck and have a nice day!
Regards,
\(\displaystyle |\pi\rangle\)

 

[MarkFL]

For the 3rd problem, I would observe that:

\(\displaystyle 11-6\sqrt{2}=3^2-2\cdot3\sqrt{2}+\sqrt{2}^2\)

Now it should be straightforward to factor this as the square of a surd.

 

[soroban]

Hello, abrk!

HallsofIvy gave you hints for the first two problems.

$$\text{3) Find: }\;\sqrt{11-6\sqrt{2}}$$

By now you may suspect that $$11-6\sqrt{2}$$ is a perfect square.

Let $$(a + b\sqrt{2})^2 \:=\:11-6\sqrt{2}$$
. . where $$a > 0$$ and $$a,b$$ are rational numbers.

Then: .$$a^2 + 2b^2 + 2\sqrt{2}ab \:=\:11-6\sqrt{2}$$

Equate coefficients: .\begin{Bmatrix}a^2+2b^2 \:=\:11 & [1] \\ 2ab \:=\:\text{-}6 & [2] \end{Bmatrix}

From [2]: .$$b \,=\,\text{-}\tfrac{3}{a}\;\;[3]$$
Substitute into [1]: .$$a^2 + 2\left(\text{-}\tfrac{3}{a}\right)^2 \:=\:11$$

$$a^4 - 11a^2 + 18 \:=\:0 \quad\Rightarrow\quad (a^2-2)(a^2-9) \:=\:0$$

. . $$a \;=\;\pm\sqrt{2},\;\pm3$$

Hence: .$$\boxed{a \,=\,3}$$

Substitute into [3]: .$$b \,=\,\text{-}\tfrac{3}{3} \quad\Rightarrow\quad \boxed{b \,=\,\text{-}1}$$Hence: .$$(3-\sqrt{2})^2 \:=\:11 - 6\sqrt{2}$$

Therefore: .$$\sqrt{11-6\sqrt{2}} \;=\;3-\sqrt{2}$$

 

[abrk]

I didn't even suspect that level of help and good will is possible nowadays.

Thank you very very much.