MHB How to distribute square roots without making common mistakes?

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The discussion focuses on the distribution of square roots in algebraic expressions, highlighting common mistakes made during the process. A participant is struggling with distributing terms correctly, specifically when dealing with negative signs in expressions. The importance of changing the signs of all terms within parentheses when a negative sign is applied is emphasized, as this is a frequent source of error. Clarifications are provided on how to simplify expressions accurately, particularly in the context of the difference of squares. Understanding these distribution rules is crucial for correctly solving algebraic problems involving square roots.
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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.
 
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Cuberoot said:
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?

[math](a+b)(a-b) = a^2-b^2[/math]
 
SuperSonic4 said:
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.
 
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?
 
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