The curious case of 50x^2 - 72

[Bavariadude]

I chose this subject title to get your attention.

Anyway, I know how to solve it; it's obviously a matter of factoring and the difference of two squares:

50x^2 - 72
2(25x^2 - 36)
2(5x - 6) (5x + 6)

However, when I try to solve it the normal way -- by factoring -- things go wrong:

50x^2 - 72
2(25x^2 - 36)
2(25x^2 - 450x + 450x - 36)

Here is where things get messy. I can't seem to factor it all the way down to 2(5x - 6) (5x + 6):

2(5x[5x - 90] +3[150x - 12])

If I go for the greatest common factor, I get this:

2(25x[x - 18] +6[75x - 6])

And if I pick a suitable factor, I get this:

2(5x[5x - 90] +6[75x - 6])

In the last one, I was able to form (5x +6), but here's where things come to a dead end.

What confuses me is that if we take a similar problem that can be solved smoothly through the difference of two squares, like, 4x^2 - 9 -- which immediately becomes (2x - 3) (2x + 3) -- it can still be solved through plain factoring:

4x^2 - 9
4x^2 - 6x + 6x - 9
2x(2x - 3) +3(2x - 3)
(2x - 3) (2x + 3)

So why is the straightforward factoring method succeeding with 4x^2 - 9, but failing with 50x^2 - 72?

 

[carrotcake10]

When you get down to the answer of

2(5x - 6) (5x + 6)

None of these factors are being added, they are being multiplied.
So when you go down the path of

2(5x[5x - 90] +3[150x - 12])

You should now be expecting an answer to have various factors being added or subtracted, which is expected.

Not all polynomials can be factored the same way. In fact, most of my math experiences there are multiple methods to solve problems. You just need to be able to look at the problem initially and try to deduce the easiest method of factorization.

 

[HallsofIvy]

I chose this subject title to get your attention.

Anyway, I know how to solve it; it's obviously a matter of factoring and the difference of two squares:

50x^2 - 72
2(25x^2 - 36)
2(5x - 6) (5x + 6)

However, when I try to solve it the normal way -- by factoring -- things go wrong:

50x^2 - 72
2(25x^2 - 36)
2(25x^2 - 450x + 450x - 36)

Why are you adding and subtracting 450x? That is, of course, 25(36) but in the example you have below, 4x^2 - 9, you add and subtract 6= 2(3) not 4(9). Since 25= 5^2 and 36= 6^3, following that same pattern, you should add and subtract 5(6)x= 30x:

2(25x^2- 30x+ 30x- 36)= 2(5x(5x- 6)+ 6(5x- 6))= 2(5x-6)(5x+6), exactly what you got before.
Actually, your first method, recognizing that (25x^2- 36) is the difference of two squares, is factoring. What you do in the second method, adding and subtracting a number times x, is a rather unusual method that only works in special situations and is NOT "the normal way".

Here is where things get messy. I can't seem to factor it all the way down to 2(5x - 6) (5x + 6):

2(5x[5x - 90] +3[150x - 12])

If I go for the greatest common factor, I get this:

2(25x[x - 18] +6[75x - 6])

And if I pick a suitable factor, I get this:

2(5x[5x - 90] +6[75x - 6])

In the last one, I was able to form (5x +6), but here's where things come to a dead end.

What confuses me is that if we take a similar problem that can be solved smoothly through the difference of two squares, like, 4x^2 - 9 -- which immediately becomes (2x - 3) (2x + 3) -- it can still be solved through plain factoring:

4x^2 - 9
4x^2 - 6x + 6x - 9
2x(2x - 3) +3(2x - 3)
(2x - 3) (2x + 3)

So why is the straightforward factoring method succeeding with 4x^2 - 9, but failing with 50x^2 - 72?

It succeeded in the second problem but not the first because you did it RIGHT in the second problem but not the first!

 

[Bavariadude]

Whoa! You're right HallsofIvy! All this time I've actually been dealing with 450 from 900/2 instead of 30, the square root of 900! I'm so embarrassed at this ridiculous mistake!

What you do in the second method, adding and subtracting a number times x, is a rather unusual method that only works in special situations and is NOT "the normal way".

I have to disagree there. It is not unusual to treat 50x^2 - 72 as ax^2 - bx - c, with b = 0. In the second example, I have gone with 4(-9). It brought me to -36, and the two numbers that add up to 0 and multiply to give -36 are -6 and +6. I've tried it several times and it worked. The only reason I've messed up with 50x^2 - 72 is because I've mistakenly went with -450 and +450 (which really do add up to 0, but multiply to -202500) instead of -30 and +30, which add up to 0 and multiply to -900, the product of 25(-36). I rushed and failed to notice the error.

Anyway, thank you both for your assistance. I've learned my lesson. Next time, I'll be very, very careful.