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In the equation x^2+bx+12=0, x is a variable and b is a

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In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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In the equation x^2+bx+12=0, x is a variable and b is a constant. What is the value of b?

(1) x-3 is a factor of x^2+bx+12.
(2) 4 is a root of the equation x^2 + bx + 12 = 0.

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/in-the-equat ... 66784.html

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Originally posted by Baten80 on 21 Feb 2011, 05:38.
Last edited by Bunuel on 23 Sep 2018, 20:34, edited 2 times in total.
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 21 Feb 2011, 06:40
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Baten80 wrote:
In the equation x^2+bx+12=0, x is variable and b is a constant. What is the value of b?
(1) x-3 is a factor of x^2+bx+12.
(2) 4 is a root of x^2+bx+12=0


You don't really need Viete's formula for the roots of a quadratic equation.

(1) x-3 is a factor of x^2+bx+12 --> simply means that x=3 is a root of x^2+bx+12=0 (if x-3 is a factor of x^2+bx+12 then x^2+bx+12=(x-3)(x-k)=0, for some k, so x=3 is on of the roots of the equation) --> substitute x=3: 3^2+3b+12=0 --> b=-7. Sufficient.

(2) 4 is a root of x^2+bx+12=0 --> the same here, substitute x=4: 4^2+4b+12=0 --> b=-7. Sufficient.

Answer: D.
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 21 Feb 2011, 06:18
3
2
x^2+bx+12=0
a, coefficient of x^2=1
b, coefficient of x=b
c, constant in the quadratic polynomial=12

Let \(\alpha,\beta\) be the two factors of the equation.

(1) x-3 is a factor of x^2+bx+12.

\(\alpha=3\)
\(\alpha\beta=\frac{c}{a}=\frac{12}{1}\)
\(\beta=\frac{12}{3}=4\)

\(\alpha+\beta=\frac{-b}{a}=\frac{-b}{1}\)
\(b=-7\)
Sufficient.

(2) 4 is a factor of x^2+bx+12=0
Same as statement 1.

\(\beta=4\)
\(\alpha\beta=\frac{c}{a}=\frac{12}{1}\)
\(\alpha=\frac{12}{4}=3\)

\(\alpha+\beta=\frac{-b}{a}=\frac{-b}{1}\)
\(b=-7\)
Sufficient.

Ans: "D"
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 21 Feb 2011, 06:56
Bunuel wrote:
Baten80 wrote:
In the equation x^2+bx+12=0, is variable and b is a constant. What is the value of b?
(1) x-3 is a factor of x^2+bx+12.
(2) 4 is a root of x^2+bx+12=0


You don't really need Viete's formula for the roots of a quadratic equation.

(1) x-3 is a factor of x^2+bx+12 --> simply means that x=3 is a root of x^2+bx+12=0 (if x-3 is a factor of x^2+bx+12 then x^2+bx+12=(x-3)(x-k)=0, for some k, so x=3 is on of the roots of the equation) --> substitute x=3: 3^2+3b+12=0 --> b=-7. Sufficient.

(2) 4 is a root of x^2+bx+12=0 --> the same here, substitute x=4: 4^2+4b+12=0 --> b=-7. Sufficient.

Answer: D.


Agree!!! This is much easier and way faster than the Viete's formula (I didn't know that's what it's called). Double thanks!!
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 21 Feb 2011, 06:59
Bunuel wrote:
Baten80 wrote:
In the equation x^2+bx+12=0, x is a variable and b is a constant. What is the value of b?
(1) x-3 is a factor of x^2+bx+12.
(2) 4 is a factor of x^2+bx+12=0


You don't really need Viete's formula for the roots of a quadratic equation.

(1) x-3 is a factor of x^2+bx+12 --> simply means that x=3 is a root of x^2+bx+12=0 (if x-3 is a factor of x^2+bx+12 then x^2+bx+12=(x-3)(x-k)=0, for some k, so x=3 is on of the roots of the equation) --> substitute x=3: 3^2+3b+12=0 --> b=-7. Sufficient.

(2) 4 is a root of x^2+bx+12=0 --> the same here, substitute x=4: 4^2+4b+12=0 --> b=-7. Sufficient.

Answer: D.


First post was wrong. This the correct post.
But I think answer is same.
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 21 Feb 2011, 07:16
We could also go by the quadratic extensiosn of the equation :
Since we know it has to have two rootsr Variables -

Statement 1 :-
X^2 - 3x-4x+12 = 0

x(x-3)-4(x-3) = 0

We have two root - X-3 , X-4 , So be has to be -3+-4 = -7 Sufficient :)

Statement 2:-

Since X = 4 it has be a root X-4, again using the same Expressions as above.

We get B=-7. Sufficient

Ans - D.
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 28 Feb 2011, 02:42
In the equation x^2+bx+12=0, x is a variable and b is a constant. What`s the value of b?

1) x-3 is a factor of x^2+bx+12
2) 4 is a root of the equation x^2 + bx + 12=0

Why (1) is sufficient?
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 28 Feb 2011, 02:50
1
For equation x^2 + bx + 12 = 0, we need to find value of b

1) says x-3 is a factor of x^2+bx+12, or x=3 is one of the roots of this quadratic equation, so 3^2+3b+12 = 0 or 21+3b = 0 or b = -7. Sufficient
2) says 4 is one of the roots so 4^2+4b+12 = 0 or 28+4b = 0 or b = -7. Again sufficient, so D

if x-a is a factor of any polynomial in x, then at x=a, the value of such a polynomial would be 0 or in other words x = a is one of the roots of such a polynomial
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 15 Dec 2012, 23:40
knew that if x-3 is a factor then 3 could be substituted to get x

but did not know the meaning of root :lol: thanks..

Bunuel,
is it helpful to memorize viete's formula?
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 03 Oct 2013, 05:49
D...
just find factors, already x-3, x-4 is another factor to get 12 and -3x-4x= -7,
this is possible becz x2+ bx +12 = 0, here b = +/-, to proof equation only is b= -7
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 24 Oct 2013, 05:53
Bunuel wrote:
Baten80 wrote:
In the equation x^2+bx+12=0, x is variable and b is a constant. What is the value of b?
(1) x-3 is a factor of x^2+bx+12.
(2) 4 is a root of x^2+bx+12=0


You don't really need Viete's formula for the roots of a quadratic equation.

(1) x-3 is a factor of x^2+bx+12 --> simply means that x=3 is a root of x^2+bx+12=0 (if x-3 is a factor of x^2+bx+12 then x^2+bx+12=(x-3)(x-k)=0, for some k, so x=3 is on of the roots of the equation) --> substitute x=3: 3^2+3b+12=0 --> b=-7. Sufficient.

(2) 4 is a root of x^2+bx+12=0 --> the same here, substitute x=4: 4^2+4b+12=0 --> b=-7. Sufficient.

Answer: D.


Im not really following your first point here.

How do you now that x=3 is a root of the equation? If the equation can be factorized in (X+2)(X+6), isn't three then also a factor from the equation?
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 24 Oct 2013, 07:49
waltiebikkiebal wrote:
Bunuel wrote:
Baten80 wrote:
In the equation x^2+bx+12=0, x is variable and b is a constant. What is the value of b?
(1) x-3 is a factor of x^2+bx+12.
(2) 4 is a root of x^2+bx+12=0


You don't really need Viete's formula for the roots of a quadratic equation.

(1) x-3 is a factor of x^2+bx+12 --> simply means that x=3 is a root of x^2+bx+12=0 (if x-3 is a factor of x^2+bx+12 then x^2+bx+12=(x-3)(x-k)=0, for some k, so x=3 is on of the roots of the equation) --> substitute x=3: 3^2+3b+12=0 --> b=-7. Sufficient.

(2) 4 is a root of x^2+bx+12=0 --> the same here, substitute x=4: 4^2+4b+12=0 --> b=-7. Sufficient.

Answer: D.


Im not really following your first point here.

How do you now that x=3 is a root of the equation? If the equation can be factorized in (X+2)(X+6), isn't three then also a factor from the equation?


If (x+2)(x+6)=0, then x-3 is NOT a factor of the quadratics. The roots of (x+2)(x+6)=0 are x=-3 and x=-6.

We are told that x-3 is a factor of x^2+bx+12, thus (x-3)(x-k)=0, thus x=3 is one of the roots.

Hope it's clear.
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 24 Oct 2013, 10:17
Thanks for your response, but I still don't get why x-3 is not a factor.
How can we now for sure that it is not a factor?
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 24 Oct 2013, 10:36
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 27 Dec 2014, 14:40
how can root and factor be the same thing? They are not necessarily the same. Am I wrong? Isn't the second option conveying a different meaning?
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 28 Dec 2014, 04:22
A tells x-3 a factor means x=3 is a solution so we get value of b...sufficient
B tells 4 is a solution ... hence sufficient.
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 10 Nov 2016, 10:22
Bunuel wrote:
Baten80 wrote:
In the equation x^2+bx+12=0, x is variable and b is a constant. What is the value of b?
(1) x-3 is a factor of x^2+bx+12.
(2) 4 is a root of x^2+bx+12=0


You don't really need Viete's formula for the roots of a quadratic equation.

(1) x-3 is a factor of x^2+bx+12 --> simply means that x=3 is a root of x^2+bx+12=0 (if x-3 is a factor of x^2+bx+12 then x^2+bx+12=(x-3)(x-k)=0, for some k, so x=3 is on of the roots of the equation) --> substitute x=3: 3^2+3b+12=0 --> b=-7. Sufficient.

(2) 4 is a root of x^2+bx+12=0 --> the same here, substitute x=4: 4^2+4b+12=0 --> b=-7. Sufficient.

Answer: D.



Hello Bunuel,

why do you conclude that 4 is a root of the equation if the statement just says it is a factor of (the statement from the page) ... My confusion is that x could be = 12 and 4 would still be a factor of the entire equation.

2) 4 is a factor of \(x^2+bx+12=0\) ...

if \(x = 12\), the equation would still be a multiple of 4 and:

\((12)^2+12b+12=0\)
\(156=-12b\)
\(-13=b\) which is not \(= -7\) and therefore it could be "not sufficient". What am I missing?


UPDATE: I looked for the question online and it seems that it has a mistake here, it shouldn't be "factor" ir should say "root". Source:

https://books.google.com.mx/books?id=ss ... 3F&f=false

Please someone correct that, otherwise it is very confusing.
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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New post 09 Mar 2017, 17:42
Baten80 wrote:
In the equation x^2+bx+12=0, x is a variable and b is a constant. What is the value of b?

(1) x-3 is a factor of x^2+bx+12.
(2) 4 is a factor of x^2+bx+12=0



my answer: D, but i guess i got here by lucky chance...

if (x-3) is a factor, then the other one will be (x-4), and b will be -7. sufficient.
i rewrote - if 4 is a factor, then (x-4) is a factor, and so is (x-3), and I can get to b=-7. i guess i was too excited with statement 1...
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Re: In the equation x^2+bx+12=0, x is a variable and b is a  [#permalink]

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Re: In the equation x^2+bx+12=0, x is a variable and b is a   [#permalink] 03 Jan 2018, 08:44
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