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Senior Manager  Joined: 25 Oct 2008
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Location: Kolkata,India
If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 61% (02:36) correct 39% (02:59) wrong based on 136 sessions

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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

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

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.
Math Expert V
Joined: 02 Sep 2009
Posts: 60627
If f(x) = 5x^2 and g(x) = x^2 + 12x + 85, what is the sum of all value  [#permalink]

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18
8
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 $$k_1+k_2=\frac{-(-4)}{1}=4$$

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Manager  Joined: 18 Aug 2009
Posts: 242
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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2
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
Manager  Joined: 15 Sep 2009
Posts: 87
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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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]

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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]

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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]

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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 _________________
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Director  Joined: 01 Feb 2011
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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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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]

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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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Joined: 02 Sep 2009
Posts: 60627
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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2
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 value  [#permalink]

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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]

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$$f(k+2) = 5(k+2)^2 = 5(k^2 + 4k + 4) = 5k^2 + 20k + 20$$
$$g(2k) = 4k^2 + 24k + 85$$

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

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 value  [#permalink]

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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]

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Manager  B
Joined: 26 Feb 2018
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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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Thanks a lot Bunuel !
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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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