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Math Expert V
Joined: 02 Sep 2009
Posts: 58340
If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 65% (02:37) correct 35% (02:48) wrong based on 409 sessions

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If $$f(x) = 5x^2 - 4$$ and $$g(x) = 3x + 1$$, then what is the sum of all the values of $$m$$ for which $$f(m - 1) = g(m^2 + 1)$$ ?

A. -10
B. -5
C. 0
D. 5
E. 10

Kudos for a correct solution.

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Math Expert V
Joined: 02 Sep 2009
Posts: 58340
Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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SOLUTION

If $$f(x) = 5x^2 - 4$$ and $$g(x) = 3x + 1$$, then what is the sum of all the values of $$m$$ for which $$f(m - 1) = g(m^2 + 1)$$ ?

A. -10
B. -5
C. 0
D. 5
E. 10

$$f(x) = 5x^2 - 4$$, then $$f(m - 1)=5(m-1)^2 - 4=5m^2-10 m+1$$;
$$g(x) = 3x + 1$$, then $$g(m^2 + 1)=3(m^2 + 1) + 1=3m^2+4$$.

Since given that $$f(m - 1) = g(m^2 + 1)$$, then $$5m^2-10 m+1=3m^2+4$$ --> $$2m^2-10m-3=0$$.

Next, 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 $$m_1+m_2=\frac{-(-10)}{2}=5$$.

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Intern  S
Joined: 24 Apr 2013
Posts: 15
Location: Russian Federation
Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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2
F(x)=5x^2-4
F(m-1)=5(m-1)^2-4
5(m^2-2m+1)-4
5m^2-10m+5-4
5m^2-10m+1

G(x)=3x+1
G(m^2+1)=3(m^2+1)+1
3m^2+3+1
3m^2+4

5m^2-10m+1=3m^2+4
2m^2-10m-3=0

D=b^2-4ac
D=100-4(-6)=100+24=124

x(1) = 10+2(root)31/4
x(2) = 10-2(root)31/4

10+2(root)31/4 + 10-2(root)31/4 = 20/4 = 5

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Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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1
The ans is 5

putting the values gives us 2m^2-10m-3=0

solving this gives us m= (10+sqrt(124))/4 and (10-sqrt(124))/4

21/4-1/4 =5 ans
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Posts: 266
Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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3
Bunuel wrote:

If $$f(x) = 5x^2 - 4$$and $$g(x) = 3x + 1$$, then what is the sum of all the values of $$m$$ for which $$f(m - 1) = g(m^2 + 1)$$ ?

A. -10
B. -5
C. 0
D. 5
E. 10

Kudos for a correct solution.

f(m-1)= 5(m-1)^2 -4
= 5(m^2+1-2m) -4
=5m^2+5-10m-4
= 5m^2-10m +1

g(m^2+1)= 3(m^2+1) +1
=3m^2+4

now f(m-1)=g(m^2+1)
5m^2-10m +1 = 3m^2+4
2m^2-10m -3=0

also, any quadratic equation can be written as
x^2-(sum of the roots)x + product of the roots

here since coeff. of m^2 is 2, therefore we will divide the whole expression by 2
and we will get, m^2-5m-3/2=0

thus sum of roots = 5

hence D
Intern  Joined: 29 Oct 2013
Posts: 18
Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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1
Substituting (m-1) and (m^2+1) in place of x in the functions f(x) and g(x) respectively,

5(m-1)^2-4=3(m^2+1)
5m^2-10m+5-4=3m^2+3+1
2m^2-10m=0
2m(m-5)=0

2m=0 and m-5=0

The roots of m are 0 and 5 and their sum is 5

Intern  Joined: 23 May 2014
Posts: 8
Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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1
Subbing (m-1) into f(x) yields 5m^2 - 10m + 1

Subbing (m^2 + 1) into g(x) yields 3m^2 + 4

Setting f(m-1) equal to g(m^2 + 1) yields
3m^2 + 1 = 5m^2 - 10m + 1

This simplifies to
0 = 2m^2 - 10m - 3

Using Viete's rules, we know that the sum of the roots x1 and x2 in ax^2 + bx + c is equal to -b/a, hence
x1 + x2 = -(-10)/2 = 5

This is the desired answer. Therefore, D.
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Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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$$f(x) = 5x^2 - 4$$

$$f(m-1) = 5 (m-1)^2 - 4$$

$$= 5(m^2 - 2m + 1) - 4$$

$$= 5m^2 - 10m + 5 - 4$$

$$= 5m^2 - 10m + 1$$ .............. (1)

g(x) = 3x+1

$$g(m^2 + 1) = 3(m^2 + 1) + 1$$

$$= 3m^2 + 4$$ ............ (2)

Equating (1) & (2)

$$5m^2 - 10m + 1 = 3m^2 + 4$$

$$2m^2 - 10m - 3 = 0$$

Sum of roots $$= \frac{-(-10)}{2} = 5$$

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Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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fra wrote:
Substituting (m-1) and (m^2+1) in place of x in the functions f(x) and g(x) respectively,

5(m-1)^2-4=3(m^2+1)
5m^2-10m+5-4=3m^2+3+1
2m^2-10m=0
2m(m-5)=0

2m=0 and m-5=0

The roots of m are 0 and 5 and their sum is 5

Equation would come up as $$2m^2 - 10m - 3$$

Answer = D, however individual roots cannot be 0 & 5
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Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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Thanks for pointing that out!
Math Expert V
Joined: 02 Sep 2009
Posts: 58340
Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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SOLUTION

If $$f(x) = 5x^2 - 4$$ and $$g(x) = 3x + 1$$, then what is the sum of all the values of $$m$$ for which $$f(m - 1) = g(m^2 + 1)$$ ?

A. -10
B. -5
C. 0
D. 5
E. 10

$$f(x) = 5x^2 - 4$$, then $$f(m - 1)=5(m-1)^2 - 4=5m^2-10 m+1$$;
$$g(x) = 3x + 1$$, then $$g(m^2 + 1)=3(m^2 + 1) + 1=3m^2+4$$.

Since given that $$f(m - 1) = g(m^2 + 1)$$, then $$5m^2-10 m+1=3m^2+4$$ --> $$2m^2-10m-3=0$$.

Next, 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 $$m_1+m_2=\frac{-(-10)}{2}=5$$.

Kudos points given to correct solutions.

Try NEW Algebra DS question.
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Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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If $$f(x) = 5x^2 - 4$$ and $$g(x) = 3x + 1$$, then what is the sum of all the values of $$m$$ for which $$f(m - 1) = g(m^2 + 1)$$ ?

A. -10
B. -5
C. 0
D. 5
E. 10

Kudos for a correct solution.

[/quote]

f(m-1) = g(m^2+1)
5(m-1)^2-4=3(m^2+1)+1
2m^2 - 10m - 3 = 0
Sum of roots is = -b/a
so 10/2 = 5
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Posts: 7
Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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f(m-1) = g(m^2+1)
5(m-1)^2-4=3(m^2+1)+1
2m^2 - 10m - 3 = 0
For equation aX^2+bX+C=0 sum of roots =(-b/a)

so here sum of roots= -(-10/2) = 5
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Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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Bunuel wrote:

If $$f(x) = 5x^2 - 4$$ and $$g(x) = 3x + 1$$, then what is the sum of all the values of $$m$$ for which $$f(m - 1) = g(m^2 + 1)$$ ?

A. -10
B. -5
C. 0
D. 5
E. 10

Kudos for a correct solution.

Pleas find attached the solution.

My opinion: it's highly unlikely to see a question based on these concepts in actual GMAT exam.

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Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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GMATinsight wrote:

Pleas find attached the solution.

My opinion: it's highly unlikely to see a question based on these concepts in actual GMAT exam.

Dear GMATInsight,

I'd like to draw you attention that there is an error in f(x) function. You have changed it from (-4) to (+4). The final quadratic equation should be:

2m^2 - 10m - 3 = 0
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Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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Mo2men wrote:
GMATinsight wrote:

Pleas find attached the solution.

My opinion: it's highly unlikely to see a question based on these concepts in actual GMAT exam.

Dear GMATInsight,

I'd like to draw you attention that there is an error in f(x) function. You have changed it from (-4) to (+4). The final quadratic equation should be:

2m^2 - 10m - 3 = 0

Thanks for notifying my mistake... However the pleasant surprise is that answer still doesn't change as the answer is dependent on the coefficients of x^2 and x...

So I hope that is ok if I accept the mistake made in first step however endorse the method to find answer (roots) of a quadratic equation. _________________
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Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum  [#permalink]

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_________________ Re: If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum   [#permalink] 05 Aug 2019, 20:05
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