In the equation x^2 + mx + n = 0, x is a variable and m and n are cons

Question

In the equation x^2 + mx + n = 0, x is a variable and m and n are constants. What is the value of n ?
 
(1) x + 2 is a factor of x^2 + mx + n
(2) The equation x^2 + mx + n = 0 has only one root

chetan2u · Answer

hi..
 Bunuel , typo in x2 + mx + n..

lets see the statements

(1) x + 2 is a factor of  x^2 + mx + n 
 the equation has two roots and 'n' is the product of these TWO roots..
Only one -2 is known
Insuff

(2) The equation  x^2 + mx + n = 0  has only one root
 this means  n = root^2 
insuff

combined

we know that the equation has one root and -2 is a root ..
so n= (-2)^2=4
sufficient
C

Bunuel · Answer

___________________
Typo edited. Thank you!

Luckisnoexcuse · Answer

(1) gives us x=-2 follows the equation
4-2m+n=0
n-2m=-4
one equation 2 variables
not sufficient

(2) only one root then m^2-4n=0
m^2 = 4n
one equation two variables
not sufficient

on combining
we have 
n-2m=-4 & m^2 = 4n

can be solved to get the values of m & n
4n=16+8m=m^2
m^2-8m-16=0
m=4 and n=4
Also since n = r1*r2
but only one root i.e. -2
n=(-2)^2 = 4
C

sahilvijay · Answer

Ans is C
see the pic for solution

prachi18oct · Answer

(1) x + 2 is a factor of x^2 + mx + n

implies that x = -2 
putting the value we get 4-2m+n = 0
insufficient

(2) The equation x^2 + mx + n = 0 has only one root 
only one root implies that the roots are equal
for equal root = -b/2a
insufficient

combining 1 & 2
roots are equal and root is -2
so -b/2a = -2
=> -m/2 = -2
m = 4
and putting that value in equation from 1 
4-2(4)+n =0
n=4
C is sufficient.

JeffTargetTestPrep · Answer

We need to determine the value of n.

Statement One Alone:
 
x + 2 is a factor of x^2 + mx + n.

Let f(x) = x^2 + mx + n. If x + 2 is a factor of x^2 + mx + n, then by the factor theorem, f(-2) = 0. That is, (-2)^2 + m(-2) + n = 0 or 4 - 2m + n = 0. However, since we don’t know the value of m, we can’t determine the value of n. Statement one alone is not sufficient to answer the question.

Statement Two Alone:
 
The equation x^2 + mx + n = 0 has only one root.

Let r be the root. Then we can express x^2 + mx + n as (x - r)^2:

x^2 + mx + n = (x - r)^2

x^2 + mx + n = x^2 - 2rx + r^2

We see that m = -2r and n = r^2. Since we don’t know the value of m or r, we can’t determine the value of n. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Since statement one states that x + 2 is a factor of x^2 + mx + n, x = -2 is a root of the equation x^2 + mx + nx = 0. Furthermore, since statement two states that there is only one root, x = -2 must be a double root. That is, x^2 + mx + n = (x + 2)^2:

x^2 + mx + n = (x + 2)^2

x^2 + mx + n = x^2 + 4x + 4

We see that m = 4 and n = 4.

Answer: C

dineshril · Answer

Given   x^2+mx+n=0  --- equ (1)

Find  'n'

Let r1 & r2 be the roots of the given Quadratic equation (1)
=> Now for a Standard Quad Equation of the form  ax^2+bx+c=0 
=> sum of the roots= \frac{-b}{a} 
 
=>  AND  product of the roots= \frac{c}{a}

=> Therefore for equ (1) with roots r1 & r2
=> r1+r2=-m
=> AND r1*r2=n
so if we know the roots of the given equation we can find 'n'

Statement 1  (x + 2) is a factor of  x^2 + mx + n 
=> so (x+2)=0 
=> OR x=-2= one of the roots
=> let r1=-2
=> Since r2 not known. Therefore Stat 1 NOT SUFFICIENT

Statement 2  The equation  x^2 + mx + n = 0  has only one root 
=> so both the roots are equal 
=> i.e r1=r2
=> n= r1^2 = r2^2 
=> since VALUE of the root not known. Therefore Stat2 NOT SUFFICIENT

BOTH  Stat 1 & 2
=> r1=-2 from stat 1
=> n= r1^2  from stat 2
=> Therefore n= (-2)^2 =4

Therefore 'C'

Thanks
Dinesh

freedom128 · Answer

Q: n the equation x^2 + mx + n = 0, x is a variable and m and n are constants. What is the value of n ?

(1) x + 2 is a factor of x^2 + mx + n 
We need to know the other factor. --> 2^2+2m+n = 0
NOT SUFFICIENT

(2) The equation x^2 + mx + n = 0 has only one root 
We don't what root that is.
Clearly NOT SUFFICIENT

Combining (1) and (2)
x^2 + mx + n = 0 has one root, which is 2.
Then (x-2)^2 = 0 --> x^2-4x+4=0 --> m=-4 and n=4
SUFFICIENT

Answer is (C)

RahulJain293 · Answer

Can someone explain the concept used in Statement 2? Or share a link to the relevant concept topic video or notes for equations with 1 root?

In the equation x^2 + mx + n = 0, x is a variable and m and n are cons

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