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

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tejal777
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If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   Updated on: 04 Mar 2018, 07:33
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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)?

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


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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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C

Originally posted by tejal777 on 29 Oct 2009, 02:38.
Last edited by Bunuel on 04 Mar 2018, 07:33, edited 2 times in total.
Renamed the topic and edited the question.
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If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   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)?

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

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 \(k_1+k_2=\frac{-(-4)}{1}=4\)

Answer: C.
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gmattokyo
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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   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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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   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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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   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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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   09 May 2011, 05:40
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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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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   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
The answer should be 4 :-D
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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   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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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   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..

plz do explain..Thanks in advance
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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   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..

plz do explain..Thanks in advance


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 value [#permalink] New post   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 value [#permalink] New post   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

\(k1 + k2 = - \frac{-4}{1} = 4\)

Answer: 4
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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   04 Mar 2018, 07:25
Bunuel ,

Can we have more this kinda questions of functions ?
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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   04 Mar 2018, 07:35
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loserunderachiever wrote:
Bunuel ,

Can we have more this kinda questions of functions ?


13. Functions



For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   04 Mar 2018, 07:46
Thanks a lot Bunuel !
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If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   10 Apr 2021, 22:09
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)?

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

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 \(k_1+k_2=\frac{-(-4)}{1}=4\)

Answer: C.


Something that continues to elude me is why we even care about the sum of the roots and product of the roots for Viete's formula. The math is straightforward enough, but what's the significance?

The only thing I can think of is if we are given one root and we need to find the other. OR the question is one like this one where they want the sum or product for some odd reason.
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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink] New post   17 Apr 2023, 05:46
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Re: If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value [#permalink]
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