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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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21 Feb 2011, 05:38
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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 factor of x^2+bx+12=0
[Reveal] Spoiler: OA

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Last edited by Baten80 on 21 Feb 2011, 06:46, edited 1 time in total.

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Re: QR:DS 62 Algebra Equation [#permalink]

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21 Feb 2011, 06:18
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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: QR:DS 62 Algebra Equation [#permalink]

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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.

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Re: QR:DS 62 Algebra Equation [#permalink]

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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.

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: QR:DS 62 Algebra Equation [#permalink]

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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.

First post was wrong. This the correct post.
But I think answer is same.
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Re: QR:DS 62 Algebra Equation [#permalink]

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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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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 ... [#permalink]

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28 Feb 2011, 02:50
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 ... [#permalink]

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28 Feb 2011, 03:09
Merging similar topics.
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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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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 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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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: QR:DS 62 Algebra Equation [#permalink]

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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.

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: QR:DS 62 Algebra Equation [#permalink]

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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.

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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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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24 Oct 2013, 10:36
waltiebikkiebal wrote:
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?

How is x-3 a factor of (x+2)(x+6)=0? I has two factors x+2 and x+6.
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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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In the equation x^2+bx+12=0, x is a variable and b is a [#permalink]

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

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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.

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:

Please someone correct that, otherwise it is very confusing.
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