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Comment on Factoring - Quadratics
This is one of the questions
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.
Question link: http://www
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
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!
Hi BalysLTU,
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!
Good question, Byefox.
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
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
Hi Jalaj,
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
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
Hi Jalaj,
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?
Question link: https://www
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
That's a very common error :-
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?
Be careful; case 3 (a = -1
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
Speaking of silly mistakes .
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
Bunuel's solution: https:/
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
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!
Thanks for the heads up!
Thanks for the heads up!
I edited my response accordingly.
Cheers,
Brent
https://gmatclub.com/forum
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. :)
Question link: https:/
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
Cheers :)
That's correct.
That's correct.
If x² + 6x + 5 = 0, then we can conclude that either x = -5 or x = -1 (but not both)
Cheers,
Brent