How many real roots does the equation x^2y+16xy+64y=0 have if y

Question

How many real roots does the equation  x^2y+16xy+64y=0  have if y < 0?

A. 0
B. 1
C. 2
D. 3
E. Infinite

Bunuel · Accepted Answer

You bet incorrectly. Please check the original post for the OA and the discussion above for solutions. The question asks for the number of values of x that satisfies the given equation if y < 0.

x^2y+16xy+64y=0

y(x+8)^2=0

Now, if y < 0 (so if  y 
eq 0 ), then the only way  y(x+8)^2=0  to hold true is if x = -8.

Answer: B.

Hope it's clear.

Skywalker18 · Answer

x^2y+16xy+64y=0 
=> y ( x^2 + 16x + 64) = 0
=> y (x+8)^2 = 0

if y<0 , then x=-8
So although there are 2 factors , they are the same x=-8 .
The equations has 1 distinct real root .
Answer B

mill · Answer

If x= -8 then the equation equals 0, and the meaning of y doesn't change that. Therefore, there is an infinite number of solutions

I bet on E

ScottTargetTestPrep · Answer

Factoring down the equation we have:

y(x^2 + 16x + 64) = 0

y(x + 8)(x + 8) = 0

y(x + 8)^2 = 0

Since y cannot be zero, x = -8, so we have 1 real root.

Answer: B

thereisaFire · Answer

Hi  Bunuel   KarishmaB

I chose Choice C because of the following reason.

I think, since (x+8)^2 is a quadratic equation, it can have 2 equal roots. x= -8 and -8
And the question is not asking for the number of DISTINCT real roots. Hence, we can say that the equation has 2 roots.

Let me know your thoughts.

KarishmaB · Answer

There is only one value ( - 8) for x that satisfies the equation. 
(x + 8)^2 is the process through which you find the values that satisfy. A quadratic needn't have two solutions. When the roots are the same, it has only one solution. 
How many values implies distinct values, otherwise what stops us from saying that there are 4 values -8, -8, -8, -8 or 10 values or 50 values etc.
Does that make sense?

thereisaFire · Answer

Hi  KarishmaB

Got your point. Anyways, we can't say that "there are 4 values or 10 values" because a quadratic eq. can have a max. of 2 roots.

This query is related to the meaning of the question-

What does the question mean when it is asking about the number of real roots?

I think, it is asking the number of (x,y) values- the case in which, the number of real roots would be infinite as there can multiple real values of y in (-8, y).
Am I missing something here?

KarishmaB · Answer

Since y is given to be less than 0, it is meant to be a constant. If I replace y by a, would you think on similar lines?

ax^2+16ax+64a=0 given a < 0
It makes sense that there is only one root, right?

Roots of a polynomial in x is the x co-ordinate of the points where it intersects/touches the x axis (assuming x is the independent variable). 
Is say (-8, -4) one of these points? No. This equation has only one root.

genericUser · Answer

the question should ask how many  distinct  solution does the equation have.

michailmuc22 · Answer

Hi, I got the correct result but was wondering whether there is some guideline for what the GMAT means when asking for the number of roots. In my understanding, (x+8)^2 yields "two" roots (the bracket has multiplicity of two). Or does the GMAT always refer to distinct values for the roots? Thanks in advance!

MASHOOR77AHAMMED · Answer

i just did the quadratic rule (B^2) - 4AC and got 256y^2 - 4*64y^2 = 0 , which means only one solution

How many real roots does the equation x^2y+16xy+64y=0 have if y < 0?

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