Help Solving an Equation: Step by Step Guide

[Bonhovis]

Hi all, I'm hoping someone here might be able to help me. I've been trying to rearrange this equation and solve it for \(\displaystyle x\) but as hard as I try, I just can't fathom how to do it.

\(\displaystyle \sqrt{x^2+16}=x+(n+\frac{1}{2})\lambda\)

I know what the equation needs be at the end, but it's all of the steps in between that escape me. If any can shed some light and run me through step by step, I'd be extremely grateful.

This is the rearranged equation:

\(\displaystyle x=\frac{16-{(n+\frac{1}{2})}^{2}\lambda^2}{2(n+\frac{1}{2})\lambda}\)

Kind regards.

 

[Prove It]

Start by squaring both sides...

 

[Bonhovis]

Sorry, I should've said that I've already got that far, moved the last \(\displaystyle x^2\) to the other side, but that's about as far as I am.

 

[MarkFL]

Squaring both sides as suggested, you should get:

\(\displaystyle x^2+16=x^2+2x\left(n+\frac{1}{2}\right)\lambda+\left(n+\frac{1}{2}\right)^2\lambda^2\)

Then, if we subtract $x^2$ from both sides, we have:

\(\displaystyle 16=2x\left(n+\frac{1}{2}\right)\lambda+\left(n+\frac{1}{2}\right)^2\lambda^2\)

Can you proceed?

 

[Bonhovis]

Many thanks for all your help. That's great, I understand it from where you left it, and how it rearranges into the final equation. The part that actually confuses me is the squaring step. I'm not sure if I understand the rules regarding the squaring the right side of the equation. Why does the right side end up with \(\displaystyle 2x\left(n+\frac{1}{2}\right)\lambda\) in addition to \(\displaystyle x^2\) and \(\displaystyle \left(n+\frac{1}{2}\right)^2\lambda^2\)?

Sorry if it seems like a stupid question. Kind regards.

 

[MarkFL]

Many thanks for all your help. That's great, I understand it from where you left it, and how it rearranges into the final equation. The part that actually confuses me is the squaring step. I'm not sure if I understand the rules regarding the squaring the right side of the equation. Why does the right side end up with \(\displaystyle 2x\left(n+\frac{1}{2}\right)\lambda\) in addition to \(\displaystyle x^2\) and \(\displaystyle \left(n+\frac{1}{2}\right)^2\lambda^2\)?

Sorry if it seems like a stupid question. Kind regards.

It is essentially an application of the formula:

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

Now, many beginning algebra students think the following is true:

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

This is such a commonly made mistake, it is referred to as "The Freshman's Dream."

Try it with some numbers though...here is the correct formula:

\(\displaystyle (3+4)^2=3^2+2\cdot3\cdot4+4^2\)

\(\displaystyle 7^2=9+24+16\)

\(\displaystyle 49=49\)

Here is "The Freshman's Dream:

\(\displaystyle (3+4)^2=3^2+4^2\)

\(\displaystyle 7^2=9+16\)

\(\displaystyle 49=25\)

Seems "The Freshman's Dream" is incorrect. :D

You can see where the correct formula comes from by FOILing:

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

 

[Bonhovis]

Thank you Mark! That makes perfect sense :) It looks like I was stuck in "the Freshman's dream!"