# Math Help - A few general questions and inquiries

1. ## A few general questions and inquiries

Hey everyone,

I just joined and have a few math questions and "inquires" about certain things regarding math!

I had a quiz recently . The first question was regarding circles:
$x^2+64$, find the radius (I'm pretty sure it was radius, if not, please feel free to correct me). Can someone explain this?

Is the product of two binomials always a trinomial? Explain please.

Can someone give me some links/information to the following math related subjects:
• Completing the Square
• Equation of a Circle
• Difference of Squares
• Dividing, multiplying exponents
• Solving negative exponents

I'll add more 'topics' as they come to mind.

Thanks for the help, greatly appreciated.
- $NineZeroFive$

2. ## Completing the Square

Say you are trying to solve the equation $x^2+4x-12=y$ and you are trying to solve for x when y=0

$x^2+4x-12=0\quad\Rightarrow\quad x^2+4x=12$
alright, now you have an equation, all you need to do is factor, the easiest way to do that would be by adding a number such that the left hand side becomes a perfect square, this number is $\left(\frac{b}{2}\right)^2$ b in this case is 4 (it's the coefficient of x) so solve...(remember to add to both sides)

$\left(\frac{b}{2}\right)^2=\left(\frac{4}{2}\right )^2=4$

$x^2+4x=12$

$x^2+4x+4=12+4$

$(x+2)^2=16$

$\sqrt{(x+2)^2}=\sqrt{16}$

$x+2=\pm4$

$x+2=4\quad x+2=-4$

$x=2\quad x=-6$

if you would like an explanation on why this works, please tell me.

~ $Q\!u\!i\!c\!k$

3. Originally Posted by Quick
Say you are trying to solve the equation $x^2+4x-12=y$ and you are trying to solve for x when y=0

$x^2+4x-12=0\quad\Rightarrow\quad x^2+4x=12$
alright, now you have an equation, all you need to do is factor, the easiest way to do that would be by adding a number such that the left hand side becomes a perfect square, this number is $\left(\frac{b}{2}\right)^2$ b in this case is 4 (it's the coefficient of x) so solve...(remember to add to both sides)

$\left(\frac{b}{2}\right)^2=\left(\frac{4}{2}\right )^2=4$

$x^2+4x=12$

$x^2+4x+4=12+4$

$(x+2)^2=16$

$\sqrt{(x+2)^2}=\sqrt{16}$

$x+2=\pm4$

$x+2=4\quad x+2=-4$

$x=2\quad x=-6$

if you would like an explanation on why this works, please tell me.

~ $Q\!u\!i\!c\!k$
Thanks for the help, an explanation would be great.

Also, would anyone know a formula relating to Quadratic Functions or Geometry (Circles) which uses $\pm$, I learned it a few months ago, just forgot what it was all about.

- $NineZeroFive$

Originally Posted by 905
Thanks for the help, an explanation would be great.
$(x+d)^2=(x+d)(x+d)=x^2+dx+dx+d^2=x^2+2dx+d^2$
the equation $x^2+2dx+d^2$ is usually written as $ax^2+bx+c$

now to find $d$ we could divide $b$ by 2, $\left(\frac{b}{2}\right)$

so now we know what $d$ equals. to find $c$ we need to square $d$, so $c=\left(\frac{b}{2}\right)^2$

now remember when we added $c=\left(\frac{b}{2}\right)^2$ to both sides in my last post? you'll notice that the left hand side didn't have $c$ so we added it on.

Originally Posted by 905
Also, would anyone know a formula relating to Quadratic Functions or Geometry (Circles) which uses $\pm$, I learned it a few months ago, just forgot what it was all about.

- $NineZeroFive$
There is a "quadratic formula" $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ that you can use to solve quadratic functions.

FYI the quadratic formula uses the "completing the square" method, watch:

your given equation: $ax^2+bx+c=0$

completing the square only works when there is no coefficient of x^2, so we divide by $a$: $x^2+\frac{b}{a}x+\frac{c}{a}=0$

subtract $\frac{c}{a}$ from both sides: $x^2+\frac{b}{a}x=-\frac{c}{a}$

complete the square: $x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2=\left (\frac{b}{2a}\right)^2-\frac{c}{a}$

factor: $\left(x+\frac{b}{2a}\right)^2=\left(\frac{b}{2a}
\right)^2-\frac{c}{a}$

simplify: $\left(x+\frac{b}{2a}\right)^2=\frac{b^2}{4a^2}-\frac{4ac}{4a^2}$

combine fractions: $\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$

find the square root of both sides: $x+\frac{b}{2a}=\pm\sqrt{\frac{b^2-4ac}{4a^2}}$

simplify: $x+\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac}}{2a}$

subtract $\frac{b}{2a}$ from both sides: $x=-\frac{b}{2a}\pm\frac{\sqrt{b^2-4ac}}{2a}$

compine fractions: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

~ $Q\!u\!i\!c\!k$

5. Thanks, exactly what I was looking for.

A few more things:

$ax^2+bx+c=0$ To solve for $c$, doesn't it have to be [/tex] on the left side? Whats the equation $ax^2+bx=c$ then? I've seen it around, don't really know what it means.

Heres another question I remember getting:
$x^2+25$ How many solutions? Is it a complete square? I'm guessing yes, since it x^2=x and 25=5. Correct?

Also, would you know the answer to this: $
x^2+64
$

6. I can't answer your questions completely until you tell me what the expressions equal.

Originally Posted by 905
Heres another question I remember getting:
$x^2+25$ How many solutions? Is it a complete square? I'm guessing yes, since it x^2=x and 25=5. Correct?
No it's not, always remember the rule $\sqrt{a+b}\neq\sqrt{a}+\sqrt{b}$

Originally Posted by 905
Also, would you know the answer to this: $
x^2+64
$
as you can see below, this is NOT a circle.
however, $y^2+x^2=64$ is.

7. Thanks, my memory's probably flawed.

You said $y=x^2+5$ is not a complete square. Can you explain why that is and how many solutions it has?

What is the equation $ax^2+bx=c$? Its the same thing as $ax^2+bx+c=0$ except we're solving for $c$ this time correct? Shouldn't $c$ be negative since its brought over to the other side?

Thanks,
-NineZeroFive

8. Originally Posted by Quick
No it's not, always remember the rule $\sqrt{a+b}\neq\sqrt{a}+\sqrt{b}$
Yes, I understand this, but:

$x^2+64$ How do I find out it it is a perfect square? Don't you square both terms and to find out if it is a perfect square?

9. You are an inquisitive lil' bugger

Originally Posted by 905
Thanks, my memory's probably flawed.

You said $y=x^2+5$ is not a complete square. Can you explain why that is and how many solutions it has?
there are no solutions, a good way to tell solutions is by using the discriminant of the equation $b^2-4ac$ if the answer is negative, there are no solutions, positive, two solutions, zero yields one solution. Let's try it...
$b^2-4ac=0^2-4(1)(5)=0-20=-20$
there are no solutions to your equation

What is the equation $ax^2+bx=c$? Its the same thing as $ax^2+bx+c=0$ except we're solving for $c$ this time correct? Shouldn't $c$ be negative since its brought over to the other side?

Thanks,
-NineZeroFive
I have never seen the equation $ax^2+bx=c$

P.S. I've enclosed an excel program I made for school, only type in boxes with a tab in them (if you hold your mouse over the box than a message will say that you can type in it)

10. Originally Posted by Quick
You are an inquisitive lil' bugger

there are no solutions, a good way to tell solutions is by using the discriminant of the equation $b^2-4ac$ if the answer is negative, there are no solutions, positive, two solutions, zero yields one solution. Let's try it...
$b^2-4ac=0^2-4(1)(5)=0-20=-20$
there are no solutions to your equation

I have never seen the equation $ax^2+bx=c$

P.S. I've enclosed an excel program I made for school, only type in boxes with a tab in them (if you hold your mouse over the box than a message will say that you can type in it)

I'm still a bit confused. You said their were two solutions, then you said there were none.

Hmmm..

EDIT: Nevermind, I think I understand the equation as well. Since $c$ is not known (whether it is positive or negative)as it a constant, it remains "postive". In a regular situation if it was positive on the right side, it's sign would change before going to the other side.

I think I understand now,

Thanks again,
-NineZeroFive

11. Originally Posted by NineZeroFive
EDIT: Nevermind, I think I understand the equation as well. Since $c$ is not known (whether it is positive or negative)as it a constant, it remains "postive". In a regular situation if it was positive on the right side, it's sign would change before going to the other side.

I think I understand now,

Thanks again,
-NineZeroFive
I actually edited that wikipedia entry just to get rid of your confusion.

~ $Q\!u\!i\!c\!k$

12. ## Rules of exponents

$x^a\times x^b=x^{(a+b)}$

$x^a\div x^b=x^{(a-b)}$

$(x^a)^b=x^{(a\times b)}$

$x^{-a}=\frac{1}{x^a}$

If you want an explanation of why these rules work, don't hesitate to ask.

~ $Q\!u\!i\!c\!k$

13. Originally Posted by Quick
$x^a\times x^b=x^{(a+b)}$

$x^a\div x^b=x^{(a-b)}$

$(x^a)^b=x^{(a\times b)}$

$x^{-a}=\frac{1}{x^a}$

If you want an explanation of why these rules work, don't hesitate to ask.

~ $Q\!u\!i\!c\!k$
Sure, why not. I know a few of those already, just dont know why.

I've learn't alot today,

Thanks again.
-NineZeroFive

14. Originally Posted by NineZeroFive
Sure, why not. I know a few of those already, just dont know why.

I've learn't alot today,

Thanks again.
-NineZeroFive
Apparently I'm a good teacher

anyway,

$x^a\times x^b=x^{(a+b)}$
let's do a sample equation...

$a^3\times a^2=a^5$

write it out: $\underbrace{\overbrace{a\times a\times a}^{a^3}\times \overbrace{a\times a}^{a^2}}_{a^5}$

next rule:
$x^a\div x^b=x^{(a-b)}$

back to our sample equation:
$a^3\div a^2=a^1$

write it out: $\frac{a^3}{a^2}=\frac{a\times a\times a}{a\times a}=\frac{\not a\times \not a\times a}{\not a\times \not a}=a^1$

next rule: $(x^a)^b=x^{(a\times b)}$

our sample equation:
$(a^3)^2=a^6$

write it out: $(a^3)^2=a^3\times a^3=\underbrace{\overbrace{a\times a\times a}^{a^3}\times \overbrace{a\times a\times a}^{a^3}}_{a^6}$

next rule: $x^{-a}=\frac{1}{x^a}$

this one is not so easy to explain, but here I go...

apply rule #2 to our sample equation:
$a^2\div a^3=a^{(2-3)}=a^{-1}$

write it out: $\frac{a\times a}{a\times a\times a}=\frac{\not a\times \not a}{\not a\times \not a\times a}=\frac{1}{a}$

that is why $a^{-1}=\frac{1}{a}$

~ $Q\!u\!i\!c\!k$