I need help with quadratics and indicies?

Aug 2012
8
0
England
Factorise: x2+3xy+2y2

I was really confused about this one because of the 2y2
at the end instead of a number so I ended up with this:

= (x + 2y2)2 - (xy - 2y2)

Because

x2+ 4xy + 4y2 - xy - 2y2

= x2 + 3xy + 2y2
But it's not properly factoried so I'm unsure about it?

The next one was factorise: 4x2+ 2x + 1

I'm confused because
1 and 1 = 2 and 2 (*2) = 4 or 0 - not 2
4 and 1 (*4 and *1) = 5 or 3 - also not 2
and neither of these combinations work so I don't know what else to do?

And the last one is solve: (1/49)-1/2
I know that it means square root of 1/1/7 because the power1/2 = square root and the - means 1 over but 1/1/7 isn't a rational number and when I try turning it into a decemal it goes on forever and because it would be 1/0.143 (rounded) its still not a rational number so I am incredibly confused and would really apreciate help..? By the way I'm not aloud to use a calculator so could you please explain without one?
 

Prove It

MHF Helper
Aug 2008
12,897
5,001
Factorise: x2+3xy+2y2

I was really confused about this one because of the 2y2
at the end instead of a number so I ended up with this:

= (x + 2y2)2 - (xy - 2y2)

Because

x2+ 4xy + 4y2 - xy - 2y2

= x2 + 3xy + 2y2
But it's not properly factoried so I'm unsure about it?

The next one was factorise: 4x2+ 2x + 1

I'm confused because
1 and 1 = 2 and 2 (*2) = 4 or 0 - not 2
4 and 1 (*4 and *1) = 5 or 3 - also not 2
and neither of these combinations work so I don't know what else to do?

And the last one is solve: (1/49)-1/2
I know that it means square root of 1/1/7 because the power1/2 = square root and the - means 1 over but 1/1/7 isn't a rational number and when I try turning it into a decemal it goes on forever and because it would be 1/0.143 (rounded) its still not a rational number so I am incredibly confused and would really apreciate help..? By the way I'm not aloud to use a calculator so could you please explain without one?
If in doubt, you can always complete the square on one of the variables...

\(\displaystyle \displaystyle \begin{align*} x^2 + 3\,x\,y + 2y^2 &= x^2 + 3\,x\,y + \left(\frac{3}{2}y\right)^2 - \left(\frac{3}{2}y\right)^2 + 2y^2 \\ &= \left(x + \frac{3}{2}y\right)^2 - \frac{9}{4}y^2 + \frac{8}{4}y^2 \\ &= \left(x + \frac{3}{2}y\right)^2 - \frac{1}{4}y^2 \\ &= \left(x + \frac{3}{2}y\right)^2 - \left(\frac{1}{2}y\right)^2 \\ &= \left(x + \frac{3}{2}y - \frac{1}{2}y\right)\left(x + \frac{3}{2}y + \frac{1}{2}y\right) \\ &= \left(x + y\right)\left(x + 2y\right) \end{align*}\)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Factorise: x2+3xy+2y2

I was really confused about this one because of the 2y2
at the end instead of a number so I ended up with this:

= (x + 2y2)2 - (xy - 2y2)
You should have (x+ 2y)2 as he first term. Squaring y2 will give you y4 that you don't want.

Because

x2+ 4xy + 4y2 - xy - 2y2

= x2 + 3xy + 2y2
But it's not properly factoried so I'm unsure about it?

The next one was factorise: 4x2+ 2x + 1

I'm confused because
1 and 1 = 2 and 2 (*2) = 4 or 0 - not 2
4 and 1 (*4 and *1) = 5 or 3 - also not 2
and neither of these combinations work so I don't know what else to do?

And the last one is solve: (1/49)-1/2
I know that it means square root of 1/1/7 because the power1/2 = square root and the - means 1 over but 1/1/7 isn't a rational number and when I try turning it into a decemal it goes on forever and because it would be 1/0.143 (rounded) its still not a rational number so I am incredibly confused and would really apreciate help..? By the way I'm not aloud to use a calculator so could you please explain without one?
 

topsquark

Forum Staff
Jan 2006
11,575
3,454
Wellsville, NY
Factorise: x2+3xy+2y2

I was really confused about this one because of the 2y2
at the end instead of a number so I ended up with this:

= (x + 2y2)2 - (xy - 2y2)

Because

x2+ 4xy + 4y2 - xy - 2y2

= x2 + 3xy + 2y2
But it's not properly factoried so I'm unsure about it?
First, that was a great way to decompose the quadratic. Kudos!

There is a way to do this that will work in most cases, but this one is easier to use the "guess method."

The coefficient on the \(\displaystyle x^2\) is 1 and the coefficient of the \(\displaystyle y^2\) is 2. So the possible factors of your expression must be of the form \(\displaystyle (x \pm y)(x \pm 2y)\) where the signs on the y terms are the same.

Now just play around with \(\displaystyle \pm\) until you get the correct answer.

If that's too hard to see then I can give you the longer (but more mechanical) way to do it. It's called the "ac" method.

-Dan
 

Prove It

MHF Helper
Aug 2008
12,897
5,001
Factorise: x2+3xy+2y2

I was really confused about this one because of the 2y2
at the end instead of a number so I ended up with this:

= (x + 2y2)2 - (xy - 2y2)

Because

x2+ 4xy + 4y2 - xy - 2y2

= x2 + 3xy + 2y2
But it's not properly factoried so I'm unsure about it?

The next one was factorise: 4x2+ 2x + 1

I'm confused because
1 and 1 = 2 and 2 (*2) = 4 or 0 - not 2
4 and 1 (*4 and *1) = 5 or 3 - also not 2
and neither of these combinations work so I don't know what else to do?

And the last one is solve: (1/49)-1/2
I know that it means square root of 1/1/7 because the power1/2 = square root and the - means 1 over but 1/1/7 isn't a rational number and when I try turning it into a decemal it goes on forever and because it would be 1/0.143 (rounded) its still not a rational number so I am incredibly confused and would really apreciate help..? By the way I'm not aloud to use a calculator so could you please explain without one?
\(\displaystyle \displaystyle \begin{align*} 4x^2 + 2x + 1 \end{align*}\) does not factorise over the reals (though it does over the complex numbers). Checking the discriminant will prove this.
 
Dec 2007
3,184
558
Ottawa, Canada
Factorise: x2+3xy+2y2
(following is "I did it my way"!):
Remove the y's:
x^2 + 3x + 2
Factor:
(x + 2)(x + 1)
Put the y's back in:
(x + 2y)(x + y)
 

topsquark

Forum Staff
Jan 2006
11,575
3,454
Wellsville, NY
And the last one is solve: (1/49)-1/2
I know that it means square root of 1/1/7 because the power1/2 = square root and the - means 1 over but 1/1/7 isn't a rational number and when I try turning it into a decemal it goes on forever and because it would be 1/0.143 (rounded) its still not a rational number so I am incredibly confused and would really apreciate help..? By the way I'm not aloud to use a calculator so could you please explain without one?
Two things that ought to help you out:
\(\displaystyle 49 = 7^2\)

and
\(\displaystyle x^{-y} = \left ( \frac{1}{x} \right )\)

To get you started...
\(\displaystyle \left ( \frac{1}{49} \right ) ^{-1/2} = \left ( \frac{1}{7^2} \right ) ^{-1/2} \)

Can you finish from here? (By the way, you did get the answer: \(\displaystyle \frac{1}{ \frac{1}{7} } = 7\) which is the correct answer.

-Dan
 
Aug 2012
8
0
England
I'm still confused on the middle one, does it just not factorise?
 
Last edited:
Aug 2012
8
0
England
I'm sorry I've just clicked to that, so saying 1/1/7 would be the same as just saying 1 * 7, just like saying 1*1/7 is the same as saying 1/7.. That makes so much sence I just didn't click lol, thank you so much :)