Find the greater of the two roots of the equation x^2 + bx + c

Question

Find the greater of the two roots of the equation  x^2 + bx + c = 0 , if both the roots are distinct and are equidistant from 4 on the number line.

(1) x is a factor of the expression  x^2 + bx + c

(2) The sum of roots of the equation  x^2 + bx + c  is 8

Archit3110 · Answer

sum of root is -b and product is c
distance is same from 4
4-(-b)= 4-c
4+b=4-c
b=-c
#1
x is a factor of the expression  x^2 + bx + c 
x^2-cx+c=0
check at x=0
c=0=b
sufficient
#2
The sum of roots of the equation  x^2 + bx + c  is 8
we get that b=-8
value of c can be -8 or 16
insufficient
option A

Gigi0707 · Answer

To Archit3110, can you help me to understand why we should regard 4-(-b) as equal as 4-c? I guess the two roots are equidistant from 4, not b or c...

Archit3110 · Answer

Two roots of the given eqn
Are -b and c
Equidistant from 4 means that thier distance from 4 will be same..

Posted from my mobile device

reed990 · Answer

I think the answer is wrong. 
Let's consider the equation can be rewritten to (x-m)(x-n)=0, then b=-(m+n), c=mn
1) x is the factor, then m or n or both is 0, let's suppose it is m, then c=0, x(x-n)=0, then n can be 0 or 8 to satisfy equidistance to 4. insufficient.
2) m+n=8, this is just the other expression of "equidistance" to 4.not sufficient.

When combining 1) and 2), 2) rules out the possibility of 0+0, so it n should be 8, we get the greater number to be 8. So C is the answer.

NoeticImbecile · Answer

Hello Archit3110 , as per my understanding, -b and c are not the roots of the equation rather sum and product of the roots, what am i missing?

Yosemite98 · Answer

Lets do some prethinking first. 
If both the roots are distinct and are equidistant from 4 on the number line -> (r1 + r2) /2 = 4 -> (r1 + r2) = 8.

1) x is a factor of the expression  x^2 + bx + c  -> this means that we can factor out x of this expresion thus the expresion can be converted to x ( x + d) for some cte d. This means that c = 0. Thus x  x^2 + bx= 0  so one solution will be x = 0. As it is equidistant from 4, the other solution will be 8 -> Sufficient

2) it gives no additional information as we already knew that from the premise.

IMO A

NoeticImbecile · Answer

Hey kindly explain this part "expresion can be converted to x ( x + d) for some cte d. This means that c = 0." and what is 'cte'?

Yosemite98 · Answer

let me try to be more clear.

with "cte" I was referring to "constant". Since x is a factor of the expresion, we can factor out x of the expresion and the only possibility to do that is for c=0 since like that we will arrive to an expression of  x^2 + bx   that we can convert in  x^2 + bx= 0 .

hope it is clear

amansoni5 · Answer

Let the roots be 
r1 and r2
Given:
r1≠r2 (distinct roots)
∣r1−4∣ = ∣r2 - 4| (both are equidistant from 4)

This means that 4 is the midpoint of the two roots.

So:

(r1+r2) / 2 = 4 ⇒
r1 + r2 = 8
This is a crucial inference from the question stem itself.

Also, from the quadratic formula:

x^2 + bx + c = 0
⇒
Sum of roots = −b,
Product of roots = c
So if 
r1+r2 = 8
⇒
−b=8
therefore, b=−8

So the stem alone tells us the sum of the roots is 8.

That already gives us one thing: b = -8.

We still need to find the greater root — but we don’t yet know what the roots are individually.

Let’s evaluate each statement.

Statement (1):
x is a factor of the expression x^2+bx+c
If 
x is a factor, then:
x^2+bx+c = x(x+d) => x^2 + dx
But this would imply 
c=0

So one of the roots is 0, and the other is −d
Let’s suppose 
x ∣ x^2 + bx + c
x is a factor of the expression, which implies that one of the roots is 0. So:

Let’s suppose:

x^2 + bx +c = x(x+d) ⇒ x^2 + dx =x^2 + bx + c
⇒ c =0, b = d

So, we conclude:
c=0
One root is 0, the other is −b

Also from the stem, we know that the sum of the roots is 8:

0 + (−b) = 8
⇒ −b = 8
⇒ b = −8
Then:

The roots are 
0 and 8

The greater root is 8

Statement (1) is sufficient.

Statement (2):
The sum of roots of the equation is 8

We already deduced from the stem that the roots are equidistant from 4 ⇒ sum is 8.

So this statement adds no new information.

Hence, Statement (2) is not sufficient.

Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Answer: (A)

Find the greater of the two roots of the equation x^2 + bx + c

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

Find the greater of the two roots of the equation x^2 + bx + c

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards