How to distribute square roots without making common mistakes?

[Cuberoot1]

I'm homeschooled, but it's gotten to the point that my Mom doesn't know how to do what she's teaching me anymore. So now I'm teaching myself with just a textbook and no one to explain it to me. I'm stuck on an issue probably simple, but I still need help. I believe I messed up on the last lines.

Please explain in detail.

#1
(sqrt7+4)(sqrt7-1) = (sqrt7+4)sqrt7-(sqrt7+4)1 = (sqrt7sqrt7+4sqrt7)-(1sqrt7+4•1) = 7+4sqrt7-sqrt7+4 = ?11+3sqrt7?

#2
(Sqrt2x+3)(sqrt2x-3) = (sqrt2x+3)sqrt2x-(sqrt2x+3)3 = (sqrt2xsqrt2x+3sqrt2x)-(3sqrt2x+3•3) = 2x+3sqrt2x-3sqrt2x+9 = ?2x+9?

I can't figure out what to add, and what to subtract on the last part.

 

[SuperSonic4]

I'm homeschooled, but it's gotten to the point that my Mom doesn't know how to do what she's teaching me anymore. So now I'm teaching myself with just a textbook and no one to explain it to me. I'm stuck on an issue probably simple, but I still need help. I believe I messed up on the last lines.

Please explain in detail.

#1
$$(\sqrt7+4)(\sqrt7-1) = (\sqrt7+4)\sqrt7-(\sqrt7+4)1 = (\sqrt7\sqrt7+4\sqrt7)-(1\sqrt7+4•1) = 7+4\sqrt7-\sqrt7+4 = ?11+3\sqrt7?$$

You're fine up until you distribute the final bracket. You have +4 and you need -4 (the minus sign means change the sign of every term in the bracket)

#2
$$(\sqrt{2x}+3)(\sqrt{2x}-3) = (\sqrt{2x}+3)\sqrt{2x}-(\sqrt{2x}+3)3 = (\sqrt{2x}\sqrt{2x}+3\sqrt{2x})-(3\sqrt{2x}+3•3) = 2x+3\sqrt{2x}-3\sqrt{2x}+9 = ?2x+9?$$

I can't figure out what to add, and what to subtract on the last part.

Are you familiar with the difference of two squares?

Spoiler

[math](a+b)(a-b) = a^2-b^2[/math]

 

[Cuberoot1]

You're fine up until you distribute the final bracket. You have +4 and you need -4 (the minus sign means change the sign of every term in the bracket)

Could you explain why I change all + to - ?

I understand the rest. Thanks.

 

[Jameson]

Hi Cuberoot,

This is a common mistake that people make or overlook when quickly doing problems. Remember that $-(a+b)=-a-b$. You have to distribute the negative to both terms inside the parentheses. Maybe it helps to think of it like this instead. $-(a+b)=-1(a+b)$?

In your problem your second step is correct, but you need to fix this part...

$$(\sqrt7\sqrt7+4\sqrt7){\color{red}-(1\sqrt7+4\cdot1)}$$? How should the part in red but simplified?