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# If f(x)=5x^2 and g(x)=x^2 + 12x + 85, what is the sum of all

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If f(x)=5x^2 and g(x)=x^2 + 12x + 85, what is the sum of all [#permalink]

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29 Oct 2009, 02:38
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If f(x)=5x^2 and g(x)=x^2 + 12x + 85, what is the sum of all values for k such that f (k+2)=g(2k)?

[Reveal] Spoiler:
Looks pretty simple..however I got stuck mid-way.
Got till $$K^2-4K-65=0$$
Does anyone know how to proceed further??

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Last edited by Bunuel on 07 Oct 2012, 04:42, edited 1 time in total.
Renamed the topic and edited the question.
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29 Oct 2009, 03:11
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29 Oct 2009, 03:28
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tejal777 wrote:
$$If f(x)=5x^2 and g(x)=x^2 + 12x + 85$$, what is the sum of all values for k such that f (k+2)=g(2k) ?

Looks pretty simple..however I got stuck mid-way.
Got till $$K^2-4K-65=0$$
Does anyone know how to proceed further??

You've done everything right:

$$5*(k+2)^2=(2k)^2+12(2k)+85$$ --> $$k^2-4k-65=0$$.

Viete's formula for the roots $$x_1$$ and $$x_2$$ of equation $$ax^2+bx+c=0$$:

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

So in our case the roots $$k_1+k_2=\frac{-(-4)}{1}=4$$
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29 Oct 2009, 03:39
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Bunuel wrote:
tejal777 wrote:
$$If f(x)=5x^2 and g(x)=x^2 + 12x + 85$$, what is the sum of all values for k such that f (k+2)=g(2k) ?

Looks pretty simple..however I got stuck mid-way.
Got till $$K^2-4K-65=0$$
Does anyone know how to proceed further??

You've done everything right:

$$5*(k+2)^2=(2k)^2+12(2k)+85$$ --> $$k^2-4k-65=0$$.

Viete's formula for the roots $$x1$$ and $$x2$$ of equation $$ax^2+bx+c=0$$:

$$x1+x2=\frac{-b}{a}$$ AND $$x1*x2=\frac{c}{a}$$

So in our case the roots $$k1+k2=\frac{-(-4)}{1}=4$$

Wow! math buster
don't have it in GMAT notes... added now, thnx and +1
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29 Oct 2009, 04:36
after simplification,we get the quadratic eqn

k^2-4k-65= 0

sum of all values of k will be equal to the sum of the roots of the above q.eqn = -b/a = 4

I will go with 4
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29 Oct 2009, 05:47
I tried solving this by quadratic equation formula and got 2 answers - 2+ or -\sqrt{65}.

What could be the other two possible values of k?
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09 May 2011, 05:40
5(k+2)^2 = 4k^2 + 24k + 85

=> 5(k^2 + 4k + 4) = 4k^2 + 24k + 85

=> 5k^2 + 20k + 20 = 4k^2 + 24k + 85

=> k^2 - 4k - 65 = 0

No need to solve this, we need sum of two values of k:

So sum of roots = -b/a = -(-4) = 4
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09 May 2011, 14:39
f(x)=5x^2
g(x)=x^2+12x+85
f(k+2)=5 (k+2)^2= 5(k^2+4k+4)=5k^2+20k+20

g(2k)= (2k)^2+ 12(2k)+85=4k^2+24k+85

5^k2+20k+20=4k^2+24k+85
k^2-4k-65=0

Using the quadratic equation (-b +/-\sqrt{b2-4ac}/2a we find the roots as follows

-(-4)+/-\sqrt{4^2 - 4(1)(-65)}/2(1)
So the first root is
4+\sqrt{16+260}/2 = 4+\sqrt{276}/2= 4+2\sqrt{69}/2
The second root is
4-\sqrt{16+260}/2 = 4-\sqrt{276}/2= 4-2\sqrt{69}/2

The sum of the roots is
4+2\sqrt{69}/2 + 4-2\sqrt{69}/2 = 8/2= 4
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09 May 2011, 16:59
sum of two roots of a quadratic equation is -b/a

now we have $$k^2-4k-65$$

=> - b/a = 4
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24 Nov 2011, 23:42
I also got 4
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07 Oct 2012, 01:38
How do we get 4??

-b/a?? what is concept behind it??

k2-4k-65=0?? i cant solve after that..

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07 Oct 2012, 04:48
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sanjoo wrote:
How do we get 4??

-b/a?? what is concept behind it??

k2-4k-65=0?? i cant solve after that..

It's called Viete's theorem, which 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 $$x_1*x_2=\frac{c}{a}$$.

Now, if we apply this to $$k^2-4k-65=0$$, we'll get that the sum of the roots must be $$k_1+k_2=\frac{-(-4)}{1}=4$$.

Hope it's clear.
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Re: If f(x)=5x^2 and g(x)=x^2 + 12x + 85, what is the sum of all [#permalink]

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07 Oct 2012, 10:59
Thanks Alot BUNUEL..

i didnt see formula like this in my life new things very hard to digest ..
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Re: If f(x)=5x^2 and g(x)=x^2 + 12x + 85, what is the sum of all [#permalink]

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10 Dec 2012, 05:13
$$f(k+2) = 5(k+2)^2 = 5(k^2 + 4k + 4) = 5k^2 + 20k + 20$$
$$g(2k) = 4k^2 + 24k + 85$$

$$5k^2 + 20k + 20 = 4k^2 + 24k + 85$$
$$k^2 - 4k -65 = 0$$

Trick: Sum of all roots: k1 + k2 = -b/a

$$k1 + k2 = - \frac{-4}{1} = 4$$

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Re: If f(x)=5x^2 and g(x)=x^2 + 12x + 85, what is the sum of all [#permalink]

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10 Jan 2014, 14:57
Hello from the GMAT Club BumpBot!

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Re: If f(x)=5x^2 and g(x)=x^2 + 12x + 85, what is the sum of all [#permalink]

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23 Apr 2014, 06:57
Could we try posting some answer choices for this one?

I suggest the following

A) -4
B) 4
C) 65
D) -8
E) 8

Thanks!
Cheers
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Re: If f(x)=5x^2 and g(x)=x^2 + 12x + 85, what is the sum of all [#permalink]

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13 Jan 2016, 10:16
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If f(x)=5x^2 and g(x)=x^2 + 12x + 85, what is the sum of all   [#permalink] 13 Jan 2016, 10:16
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