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Retired Moderator B
Joined: 27 Aug 2012
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If x^2 + 12x − k = 0, is x = 4?  [#permalink]

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10 00:00

Difficulty:   75% (hard)

Question Stats: 53% (02:01) correct 47% (02:07) wrong based on 235 sessions

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If x^2 + 12x − k = 0, is x = 4?

(1) (x + 16) is a factor of x^2 + 12x − k = 0, where k is a constant, and x is a variable.

(2) x ≠ −16

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Re: If x^2 + 12x − k = 0, is x = 4?  [#permalink]

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If x^2+12x−k=0, is x=4?

(1) (x+16) is a factor of x^2+12x−k=0, where k is a constant, and x is a variable. This implies that we can factor out $$x+16$$ from $$x^2+12x-k=0$$, so we would have $$(x+16)*(something)=0$$. Thus x=-16 is one of the roots of the given quadratic equation.

Viete's theorem states that for the roots $$x_1$$ and $$x_2$$ of a quadratic equation $$ax^2+bx+c=0$$:

$$x_1+x_2=\frac{-b}{a}$$ AND $$x_1*x_2=\frac{c}{a}$$.

Thus according to the above $$x_1+x_2=-16+x_2=\frac{-12}{1}$$ --> $$x_2=4$$.

So, we have that x is either -16 or 4. Not sufficient.

(2) x≠−16. Clearly insufficient.

(1)+(2) Since (2) says that x is NOT -16, then x=4. Sufficient.

Hope it's clear.
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Retired Moderator B
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Re: If x^2 + 12x − k = 0, is x = 4?  [#permalink]

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Bunuel wrote:
If x^2+12x−k=0, is x=4?

(1) (x+16) is a factor of x^2+12x−k=0, where k is a constant, and x is a variable. This implies that we can factor out $$x+16$$ from $$x^2+12x-k=0$$, so we would have $$(x+16)*(something)=0$$. Thus x=-16 is one of the roots of the given quadratic equation.

Viete's theorem states that for the roots $$x_1$$ and $$x_2$$ of a quadratic equation $$ax^2+bx+c=0$$:

$$x_1+x_2=\frac{-b}{a}$$ AND $$x_1*x_2=\frac{c}{a}$$.

Thus according to the above $$x_1+x_2=-16+x_2=\frac{-12}{1}$$ --> $$x_2=4$$.

So, we have that x is either -16 or 4. Not sufficient.

(2) x≠−16. Clearly insufficient.

(1)+(2) Since (2) says that x is NOT -16, then x=4. Sufficient.

Hope it's clear.

(1)+(2)=> 1 says x=-16 and 2 says x is NOT -16...So,isn't it contradicting hence Insufficient..?

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Re: If x^2 + 12x − k = 0, is x = 4?  [#permalink]

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debayan222 wrote:
Bunuel wrote:
If x^2+12x−k=0, is x=4?

(1) (x+16) is a factor of x^2+12x−k=0, where k is a constant, and x is a variable. This implies that we can factor out $$x+16$$ from $$x^2+12x-k=0$$, so we would have $$(x+16)*(something)=0$$. Thus x=-16 is one of the roots of the given quadratic equation.

Viete's theorem states that for the roots $$x_1$$ and $$x_2$$ of a quadratic equation $$ax^2+bx+c=0$$:

$$x_1+x_2=\frac{-b}{a}$$ AND $$x_1*x_2=\frac{c}{a}$$.

Thus according to the above $$x_1+x_2=-16+x_2=\frac{-12}{1}$$ --> $$x_2=4$$.

So, we have that x is either -16 or 4. Not sufficient.

(2) x≠−16. Clearly insufficient.

(1)+(2) Since (2) says that x is NOT -16, then x=4. Sufficient.

Hope it's clear.

(1)+(2)=> 1 says x=-16 and 2 says x is NOT -16...So,isn't it contradicting hence Insufficient..?

(1) says that x=-16 OR x=4. The equation is $$x^2+12x-64=0$$ (k=64) --> x=-16 OR x=4.
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Re: If x^2 + 12x − k = 0, is x = 4?  [#permalink]

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Got it!

I was doing the mistake by considering the Stat.1 ONLY not focusing on the solns of the eqn in 1...!

Thanks for clarifying.
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Re: If x^2 + 12x − k = 0, is x = 4?  [#permalink]

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Bunuel wrote:
If x^2+12x−k=0, is x=4?

(1) (x+16) is a factor of x^2+12x−k=0, where k is a constant, and x is a variable. This implies that we can factor out $$x+16$$ from $$x^2+12x-k=0$$, so we would have $$(x+16)*(something)=0$$. Thus x=-16 is one of the roots of the given quadratic equation.

Viete's theorem states that for the roots $$x_1$$ and $$x_2$$ of a quadratic equation $$ax^2+bx+c=0$$:

$$x_1+x_2=\frac{-b}{a}$$ AND $$x_1*x_2=\frac{c}{a}$$.

Thus according to the above $$x_1+x_2=-16+x_2=\frac{-12}{1}$$ --> $$x_2=4$$.

So, we have that x is either -16 or 4. Not sufficient.

(2) x≠−16. Clearly insufficient.

(1)+(2) Since (2) says that x is NOT -16, then x=4. Sufficient.

Hope it's clear.

Why A is not sufficient? X cannot be -16, as in this case X+16 =0. As I know 0 cannot be a factor or the integer. Pls. explan
Math Expert V
Joined: 02 Sep 2009
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Re: If x^2 + 12x − k = 0, is x = 4?  [#permalink]

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Dmitriy wrote:
Bunuel wrote:
If x^2+12x−k=0, is x=4?

(1) (x+16) is a factor of x^2+12x−k=0, where k is a constant, and x is a variable. This implies that we can factor out $$x+16$$ from $$x^2+12x-k=0$$, so we would have $$(x+16)*(something)=0$$. Thus x=-16 is one of the roots of the given quadratic equation.

Viete's theorem states that for the roots $$x_1$$ and $$x_2$$ of a quadratic equation $$ax^2+bx+c=0$$:

$$x_1+x_2=\frac{-b}{a}$$ AND $$x_1*x_2=\frac{c}{a}$$.

Thus according to the above $$x_1+x_2=-16+x_2=\frac{-12}{1}$$ --> $$x_2=4$$.

So, we have that x is either -16 or 4. Not sufficient.

(2) x≠−16. Clearly insufficient.

(1)+(2) Since (2) says that x is NOT -16, then x=4. Sufficient.

Hope it's clear.

Why A is not sufficient? X cannot be -16, as in this case X+16 =0. As I know 0 cannot be a factor or the integer. Pls. explan

Factor of a number and factor of an expression are two different things. For example, both (x+16) and (x-4) are factors of x^2+12x−64=0 --> (x+16)(x-4)=0.

Hope this helps.
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Re: If x^2 + 12x − k = 0, is x = 4?  [#permalink]

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Bunuel wrote:
Dmitriy wrote:
Bunuel wrote:
If x^2+12x−k=0, is x=4?

(1) (x+16) is a factor of x^2+12x−k=0, where k is a constant, and x is a variable. This implies that we can factor out $$x+16$$ from $$x^2+12x-k=0$$, so we would have $$(x+16)*(something)=0$$. Thus x=-16 is one of the roots of the given quadratic equation.

Viete's theorem states that for the roots $$x_1$$ and $$x_2$$ of a quadratic equation $$ax^2+bx+c=0$$:

$$x_1+x_2=\frac{-b}{a}$$ AND $$x_1*x_2=\frac{c}{a}$$.

Thus according to the above $$x_1+x_2=-16+x_2=\frac{-12}{1}$$ --> $$x_2=4$$.

So, we have that x is either -16 or 4. Not sufficient.

(2) x≠−16. Clearly insufficient.

(1)+(2) Since (2) says that x is NOT -16, then x=4. Sufficient.

Hope it's clear.

Why A is not sufficient? X cannot be -16, as in this case X+16 =0. As I know 0 cannot be a factor or the integer. Pls. explan

Factor of a number and factor of an expression are two different things. For example, both (x+16) and (x-4) are factors of x^2+12x−64=0 --> (x+16)(x-4)=0.

Hope this helps.

Thanks. I didnt know about factors of an expression.
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Re: If x^2 + 12x − k = 0, is x = 4?  [#permalink]

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