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?
How many real roots does the equation . . .
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Based on the answer, it looks like you're really asking us to solve for x.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
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?
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Brent
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We could factor as follows:
x²y + 16xy + 64y = 0
y * (x² + 16x + 64) = 0
y * (x + 8)² = 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.)
x²y + 16xy + 64y = 0
y * (x² + 16x + 64) = 0
y * (x + 8)² = 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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Is x the only unknown? All we "know" about y is that it's negative.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.)
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
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Factoring down the equation we have: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
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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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.
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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Hi Vincen,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
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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