Lesson: Factoring - Quadratics

Comment on Factoring - Quadratics

This is one of the questions below the video.

Which of the following equations has
a root in common with x^2 - 6x + 5 =
0?
(A) x^2 + 1 = 0
(B) x^2 - x - 2 =0
(C) x^2 - 10x - 5 =0
(D) 2x^2 - 2 =0
(E) x^2 - 2x - 3 =0

I understood the solution but I have a question:

How X^2+1=0? Is this even possible? X^2 could be either zero or a positive number.
gmat-admin's picture

Question link: http://www.beatthegmat.com/common-root-t117001.html

Great question, Mohammad!

Not all equations have a solution. This equation (x^2 + 1 = 0) is one of them.

ASIDE: some students with an understanding of complex (aka imaginary) numbers, will say that there IS a solution. It's x = i (where i = √-1). However, the GMAT only covers REAL numbers, so we say that the equation x^2 + 1 = 0 has no REAL solution.

Thanks. And great point on the imaginary numbers.

Hi Brent,

Great video! Just a quick question - how does one go about factoring a quadratic equations such as these ones:

1) 2x^2 + 9x + 9 = 0
This one factors out to be (2x + 3) (x + 3)
2) 2x^2 + x - 3
Factors out to (x - 1) (2x + 3)

I'm confused on how to separate factors such as 2x in individual expressions in the above examples.

Hope this question made sense!

gmat-admin's picture

Hi BalysLTU,

There's a formal technique for factoring quadratics where the coefficient of the x² is NOT 1, but for the purposes of the GMAT, we can typically apply the informal method describe below.

Here's a video of the formal technique: https://www.khanacademy.org/math/algebra/polynomial-factorization/factor...

On the GMAT, you'll typically see quadratic equations that look like the following:
x² - 2x - 15 = 0
x² + 8x + 12 = 0
x² - 7x + 10 = 0

However, if you do encounter a quadratic equation where the coefficient of the x² is NOT 1, then you can typically factor the expression by applying some number sense and testing some values.

Here's what I mean:
Given: 2x² + 9x + 9 = 0
Let's say the expression (2x² + 9x + 9) can be factored to look something like: (a + b)(c + d)
What can we conclude about some of the values?

Well, we know that ac = 2x² (applying the FOIL method)
If we limit ourselves to integer vales, then there's only one way to get a product of 2x²

That is, (2x)(x) = 2x²

So, we already know that 2x² + 9x + 9 = (2x + b)(x + d)

We also know that bd = 9
There aren't that many options where the product of two integers equals 9
One option is (1)(9) = 9
Another option is (3)(3) = 9

At this point, we can start TESTING some options...
How about b = 1 and d = 9
Plug in to get: (2x + 1)(x + 9)
Expand and simplify to get: (2x + 1)(x + 9) = 2x² + 19x + 9
The middle term (19x) is not right. We want 9x

TRY AGAIN

How about b = 3 and d = 3
Plug in to get: (2x + 3)(x + 3)
Expand and simplify to get: (2x + 3)(x + 3) = 2x² + 9x + 9
Perfect!
Done!

I'll leave it to you to try factoring the second expression.

Cheers,
Brent

Hey!
How do you factorize x² - 10x - 5? I cant seem to think of two numbers that will add to be - 10 and multiply to be 5. Is it because it consists of two negatives? (Since its not x² + nx + p). I'm confused what to do when both the x term and the constant are negative, as opposed to two positives or one positive and one negative. Cheers!
gmat-admin's picture

Good question, Byefox.

x² - 10x - 5 cannot be factored into the form (x + ?)(x - ?)

Can you tell me the details of the question that require the factorization of x² - 10x - 5? Perhaps there's an alternate approach.

Cheers,
Brent

Hi Brent, for the following question my answer is D (both sufficient)

If xy > 0, does (x - 1)(y - 1) = 1?

(1) x + y = xy
(2) x = y

statement is: x+y=xy
1) - clearly sufficient
2) - x^2=2x >>>> x=2

But the correct answer given is A. Can't figure why!

https://gmatclub.com/forum/if-xy-0-does-x-1-y-167337.html
gmat-admin's picture

Hi Jalaj,

Here's my step-by-step solution: https://gmatclub.com/forum/if-xy-0-does-x-1-y-167337.html#p2112935

Cheers,
Brent

Hi Brent, could you provide a simpler way to define the question and the answer?

Which of the following equations has a root in common with x² − 6x + 5 = 0?
A. x² + 1 = 0
B. x² − x − 2 = 0
C. 2x² − 2 = 0
D. x² − 2x − 3 = 0
E. x² − 10x − 5 = 0
gmat-admin's picture

Hi Jalaj,

I have a solution here: https://gmatclub.com/forum/which-of-the-following-equations-has-a-root-i...

Please let me know if you'd like me to clarify any steps.

Cheers,
Brent

Hi Brent:

In the following question: https://www.beatthegmat.com/what-is-the-value-of-r-2-2rs-s-2-t297050.html

What is the value of r^2-2rs+s^2?

(1) s=4
(2) r-s=12

Why is statement 1 not sufficient?

We know that r^-2rs+s^2 is the same as (r-s)(r-s).
If we are given s=4.
Then our quadratic equation becomes (r-4)(r-s)=0. This gives us a definitive answer of r=4.

Could you kindly explain what I am overlooking?
gmat-admin's picture

Question link: https://www.beatthegmat.com/what-is-the-value-of-r-2-2rs-s-2-t297050.html

For some reason, you have turned the EXPRESSION (r-s)(r-s) into an EQUATION (r-s)(r-s) = 0
There is no equation here.
We're asked to determine the value of (r-s)(r-s), but you are trying to find the value of r.

So, if s = 4, then (r - s)(r - s) = (r - 4)(r - 4)
Since we don't know the value of r, we can't find the value of (r - 4)(r - 4)

Does that help?

Cheers,
Brent

Thanks Brent, you pointed out the exact mistake that I had made and overlooked. I should not have assumed that r^2-2rs+s^2 is an equation. This makes sense now, thanks!
gmat-admin's picture

That's a very common error :-)

Hi Brent,

https://gmatclub.com/forum/if-ax-b-0-is-x-0-1-a-b-0-2-a-b-99749.html

Need your approach to answer this one. I used a very lengthy approach and still got the answer wrong.

ax + b = 0 > Case a. ax = -2 and b = 2 or Case b. ax = 2 and b = -2.
Case a can have 4 scenarios:ax = -2 and b = 2
1. a = 2 and x = -1
2. a = -2 and x = 1
3. a = -1 and x = 2
4. a = 1 and x = -2

Snt 1: a+b>0; only 1 and 4 fits and hence sufficient.

Used the same method for Snt 2. Could you help me identify where did I go wrong?
gmat-admin's picture

Be careful; case 3 (a = -1 and x = 2) also meets the condition that a+b > 0

Cheers,
Brent

Oh no, yet again! Thanks for pointing out Breant, I have been making such silly mistakes a lot.
gmat-admin's picture

Speaking of silly mistakes . . . . :-)

Hi Brent,

I wanted to more clarity on the or condition shared here by Bunuel - https://gmatclub.com/forum/what-is-the-value-of-x-y-163356.html#p1937355

Please share more examples to understand the concept. I marked D considering B is also sufficient.

Warm Regards,
Pritish
gmat-admin's picture

Bunuel's solution: https://gmatclub.com/forum/what-is-the-value-of-x-y-163356.html#p1937355

This question highlights an important concept: If AB = 0, then EITHER A = 0 OR B = 0
For example, if AB = 0, then it COULD be the case that A = 0 and B = 1
Or it COULD be the case that A = 3 and B = 0
Or it COULD be the case that A = 0 and B = 0

Many students will incorrectly conclude that, if AB = 0, then A and B must BOTH equal zero. As you can see by the above examples, this is not necessarily true.

Consider this quadratic equation: x² + 2x - 15 = 0
Factor to get: (x + 5)(x - 3) = 0
This means that EITHER x + 5 = 0 OR x - 3 = 0
So, EITHER x = -5 OR x = 3
So, if a DS question asked "What is the value of x?" and statement 1 was x² + 2x - 15 = 0, then statement 1 is NOT sufficient (since x can be either -5 or 3)

Most of the linked questions DS in the Reinforcement Activities box at https://www.gmatprepnow.com/module/gmat-algebra-and-equation-solving/vid... will feature a similar feature.

For example:
- https://gmatclub.com/forum/if-y-x-2-6x-9-what-is-the-value-of-x-221249.html
- https://gmatclub.com/forum/what-is-the-value-of-integer-n-243142.html
- https://gmatclub.com/forum/is-y-214659.html

Cheers,
Brent

https://gmatclub.com/forum/for-integers-x-and-y-which-of-the-following-must-be-an-integer-244274.html

Hi Brent! Just a small error. Here we can eliminate option A since it's actually √91 .. I think you put √81 by mistake.
Amazing videos btw. Thank you!
gmat-admin's picture

Thanks for the heads up!
I edited my response accordingly.

Cheers,
Brent

https://gmatclub.com/forum/what-is-the-value-of-x-y-163356.html

Hi Brent. Couldn't quite understand Bunuel's solution for statement 2. I put D, but apparently statement 2 is not sufficient. Please could you clarify?
Thanks before hand. :)
gmat-admin's picture

Question link: https://gmatclub.com/forum/what-is-the-value-of-x-y-163356.html

TARGET QUESTION: What is the value of x - y?

Statement 2: x² - y² = 0
Factor: (x + y)(x - y) = 0
So, EITHER x + y = 0, OR x - y = 0
So, x - y may or may not equal 0
Not sufficient.

Alternatively, we could test some cases.

case a) x = 1 and y = -1 (this satisfies the condition that x² - y² = 0)
In this case, the answer to the target question is x - y = 2

case b) x = 1 and y = 1
In this case, the answer to the target question is x - y = 0
Not sufficient.

Cheers,
Brent

So, this EITHER OR rule works even for x^2 +6x+5=0? Would this quadratic equation mean that X=-5 OR X=-1 and not BOTH?
Cheers :)
gmat-admin's picture

That's correct.
If x² + 6x + 5 = 0, then we can conclude that either x = -5 or x = -1 (but not both)

Cheers,
Brent

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