Putting sqaure roots into an equation

[gonelli]

This is a idea that came to me when my math teacher was stressing the point that we need to write that the sqaure root of 25 is 5 and -5, and that we must include that even when that answer can only be positive.

 

So i was think that having the sqaure root of a number like 25 in an equation for a line would result in essentail two different lines. I don't really know if you're allowed to do something like that.

 

For example:

x^2 + 2x + 25^(1/2)

 

So that could be either + 5 or - 5 as a constant, right? Please let me know if this makes any sense.

[the tree]

Not really, [math]\sqrt{25}=5[/math] and that's it. "Square root", being a well defined function has only one output per input.

 

Its relationships like [math]x^{2}=25[/math] that have two solutions, namely [math]x=\pm 5[/math].

[Country Boy]

This is a idea that came to me when my math teacher was stressing the point that we need to write that the sqaure root of 25 is 5 and -5, and that we must include that even when that answer can only be positive.

I hope your teacher DIDN'T say precisely that! [math]\sqrt{25}= 5[/math], not -5. What is true is that the equation, [math]x^2= a[/math] has TWO solutions. One is [math]\sqrt{a}[/math], the other is [math]-\sqrt{a}[/math]. The reason we need to write the "-" (or sometimes [math]\pm[/math]) is BECAUSE [math]\sqrt{a}[/math] does NOT include the negative. You might then, on the basis of other information (whether a negative number is physically possible if it is a physic problem, the fact that "distance" is always positive) decide that one of those solutions does not actually work in that particular problem.

 

So i was think that having the sqaure root of a number like 25 in an equation for a line would result in essentail two different lines. I don't really know if you're allowed to do something like that.

 

For example:

x^2 + 2x + 25^(1/2)

 

So that could be either + 5 or - 5 as a constant, right? Please let me know if this makes any sense.

No. [math]25^{\frac{1}{2}}[/math] or [math]\sqrt{25}[/math] are defined as POSITIVE 5. In general, for any positive number, a, [math]a^{\frac{1}{2}}[/math] or [math]\sqrt{a}[/math] are defined as the POSITIVE number whose square is a.

[gonelli]

Ahhh, that makes sense. And by the way, my teacher was actually refering to a question where it came down to x^2 = 25, so I now see how it works. Thanks!

[the tree]

Quadratics, which are equations like that that have no higher power than 2 (in general: [math]ax^2 + bx + c = 0[/math]) always have two solutions.

 

In some cases the two solutions will be the same, in some cases different, and in some not real. (the last is for a later date)

[Country Boy]

Quadratics, which are equations like that that have no higher power than 2 (in general: [math]ax^2 + bx + c = 0[/math]) always have two solutions.

 

In some cases the two solutions will be the same, in some cases different, and in some not real. (the last is for a later date)

I would say that a quadratic equation is an equation whose highest power IS 2 (not "no higher than 2- that would include linear equations.).