How many real roots does the equation . . .

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Wed Sep 27, 2017 6:33 pm

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

The OA is B.

How can I determine it without knowing the value of y?

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Source: Beat The GMAT — Problem Solving | Wed Sep 27, 2017 6:33 pm

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Sep 28, 2017 7:17 am

Vincen wrote: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

Based on the answer, it looks like you're really asking us to solve for x.

Typically, the solution to an equation with 2 variables is an ordered pair of values (x,y) that satisfy the equation.
So, for example x = -8 and y = -1 is a solution to the given equation.
Likewise, x = -8 and y = -3 is a solution
...and (-8, -6) is a solution
...and (-8, -20) is a solution
Etc.

This question might be someone ambiguous to be a true GMAT question.
What's the source?

Cheers,
Brent

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

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by Matt@VeritasPrep » Thu Sep 28, 2017 4:18 pm

We could factor as follows:

xÂ²y + 16xy + 64y = 0

y * (xÂ² + 16x + 64) = 0

y * (x + 8)Â² = 0

Since y < 0, x must = -8. Since the equation is presumably a quadratic in x, we'd have one real root x. (I think the idea is that y is some known negative and that x is the only unknown, but the question is poorly phrased.)

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Sep 28, 2017 5:07 pm

Matt@VeritasPrep wrote:I think the idea is that y is some known negative and that x is the only unknown, but the question is poorly phrased.)

Is x the only unknown? All we "know" about y is that it's negative.

I'm wondering if it's even okay to use the word "root" when there are 2 or more variables involved.

What do you think, Matt?

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8137; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Thu Jan 18, 2018 7:58 am

Vincen wrote: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

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

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DrMaths: Senior | Next Rank: 100 Posts; Posts: 82; Joined: Mon Jan 15, 2018 2:01 am

by DrMaths » Thu Jan 18, 2018 8:22 am

The value of y is irrelevant as it doesn't affect the rest (x^2+16x+64) which must equal zero anyway (as anything times zero must be zero).
Therefore the question can be simplified to
How many real roots does the equation x^2+16x+64=0 ?

For any quadratic equation in the form ax^2+bx+c=0, we can determine the number of roots from the
Discriminant D = b^2 - 4ac
If D < 0, there are NO real roots
If D = 0, there is 1 real root
If D >0, there are 2 real roots

From x^2+16x+64=0 , we see that a = 1, b = 16 and c = 64
Therefore D = 16^2 -4*64 = 0
Hence there is exactly 1 real root.

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EconomistGMATTutor: GMAT Instructor; Posts: 555; Joined: Wed Oct 04, 2017 4:18 pmFollowed by: 12 members; Thanked: 180 times

by EconomistGMATTutor » Thu Jan 18, 2018 10:33 am

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

Hi Vincen,
Let's take a look at your question.

$$x^2y+16xy+64y=0$$
$$y\left(x^2+16x+64\right)=0$$
$$x^2+16x+64=0$$

To find the number of roots of a quadratic equation, we use the discriminant.

If Discriminant < 0, there are two distinct complex roots of the quadratic equation.
If Discriminant = 0, there is only one real root of the quadratic equation.
If Discriminant > 0, there are two distinct real roots of the quadratic equation.

So let's find the discriminant of the given equation.
$$Discriminant=b^2-4ac$$
$$=\left(16\right)^2-4\left(1\right)\left(64\right)=256-256=0$$

Since, discriminant is zero, therefore, the given equation has only one root.
Therefore, Option B is correct.

Hope it helps.
I am available if you'd like any follow up.

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