If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum

Question

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

Bunuel · Accepted Answer

Official 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

Since  f(x) = 5x^2 - 4 , then  f(m - 1)=5(m-1)^2 - 4=5m^2-10 m+1 ;

Since  g(x) = 3x + 1 , then  g(m^2 + 1)=3(m^2 + 1) + 1=3m^2+4 .

We are given that  f(m - 1) = g(m^2 + 1) , hence we can equate the two expressions:  5m^2-10 m+1 = 3m^2+4 . Simplifying this equation yields  2m^2-10m-3=0 .

Next, we can apply  Viete's theorem  which states that for the roots  x_1  and  x_2  of a quadratic equation  ax^2+bx+c=0 , the following relationships hold:

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

Thus, according to the theorem,  m_1+m_2=\frac{-(-10)}{2}=5 .

Answer: D

Try NEW Algebra  DS question .

VladimirKarpov · Answer

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

Answer is D

manish2014 · Answer

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

manpreetsingh86 · Answer

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

fra · Answer

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

Answer D

dbaremberg · Answer

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.

PareshGmat · Answer

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

Answer = D

PareshGmat · Answer

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

Answer = D,   however individual roots cannot be 0 & 5

fra · Answer

Thanks for pointing that out!

M3sSiaH · Answer

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) = 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

UttamGiet · Answer

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

GMATinsight · Answer

Pleas find attached the solution.

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

Posted from my mobile device

Mo2men · Answer

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

GMATinsight · Answer

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.

GmatPoint · Answer

Given  f\left(x\right)\ =\ 5x^2-4    g\left(x\right)\ =\ 3x+1 
﻿Substituting the function for 
 ﻿f\left(m-1\right)\ =\ 5\left(m-1\right)^2-4  =  5\left(m^2-2m+1\right)-4 
﻿=  5m^2-10m+1 
﻿Similarly for  g\left(m^2+1\right) 
﻿=  3\left(m^2+1\right)+1 
﻿Equating the two we have
 ﻿2m^2-10m-3\ =\ 0 
﻿The sum of possible values of m is equal to the sum of roots of the equation which is equal to 5.

Adesewaa · Answer

This question falls under what topic please?

Bunuel · Answer

Check the category tags, the question belongs to Algebra and Functions and Custom Characters category: https://gmatclub.com/forum/download/file.php?id=84240 GMAT-Club-Forum-7250mo5k.png

If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum

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If f(x) = 5x^2 - 4 and g(x) = 3x + 1, then what is the sum

Prep Toolkit

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My Rewards