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The equation \(x^2 + ax - b = 0\) has equal roots, and one of the roots of the equation \(x^2 + ax + 15 = 0\) is 3. What is the value of b?

A. \(-\frac{1}{64}\)
B. \(-\frac{1}{16}\)
C. \(-15\)
D. \(-16\)
E. \(-64\)
[Reveal] Spoiler: OA

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Official Solution:

The equation \(x^2 + ax - b = 0\) has equal roots, and one of the roots of the equation \(x^2 + ax + 15 = 0\) is 3. What is the value of b?

A. \(-\frac{1}{64}\)
B. \(-\frac{1}{16}\)
C. \(-15\)
D. \(-16\)
E. \(-64\)


Since one of the roots of the equation \(x^2 + ax + 15 = 0\) is 3, then substituting we'll get: \(3^2+3a+15=0\). Solving for \(a\) gives \(a=-8\).

Substitute \(a=-8\) in the first equation: \(x^2-8x-b=0\).

Now, we know that it has equal roots thus its discriminant must equal to zero: \(d=(-8)^2+4b=0\). Solving for \(b\) gives \(b=-16\).


Answer: D
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Re: D01-41 [#permalink]

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New post 21 Nov 2014, 04:21
i didn't understand the question. What does equal roots mean?
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Re: D01-41 [#permalink]

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New post 15 Feb 2015, 07:56
Hi Bunuel I have a doubt. can anyone help me understand why is the below solution wrong

x*x +a*x + 15=0 --> x*x +a*x=-15

Substituting the above value in below equation we have
x*x+ ax- b= 0 --> -15-b=0 --> b=-15

Thus answer is C@

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New post 16 Feb 2015, 04:45
qw1981 wrote:
Hi Bunuel I have a doubt. can anyone help me understand why is the below solution wrong

x*x +a*x + 15=0 --> x*x +a*x=-15

Substituting the above value in below equation we have
x*x+ ax- b= 0 --> -15-b=0 --> b=-15

Thus answer is C@


If b = -15, does x^2 - 8x - b = x^2 - 8x + 15 =0 have equal roots?
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Re: D01-41 [#permalink]

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New post 16 Feb 2015, 08:57
Bunuel after spending 30 minutes pondering where I went wrong just realized that if I chose b=-15 then it becomes the same equation as the second one. In quadratic equations if the constant value changes the root changes. Was taught these things in 8th/9th grade.. This is really embarrassing :oops: .. Thanks for replying though atleast my concept got cleared.

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I reasoned through this problem a bit different, but it seems like it landed at the right answer. With a root as x-3 and b = 15 for the second equation, you can get to x-5 (can't see another way to get a positive 15) and as a result -8 for a. With two same roots and -8 as a, you know it's (x-some number)^2, and -4 is the only number that fits. (X-4)^2 results in X2 - 8x + 16 and as a result -16 fits.

I'm able to follow along Bunel's solution up to the discriminant point; not sure what a discriminant is!

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New post 21 Mar 2015, 10:23
Substitute a=-8 in the first equation: x^2-8x-b=0.

Now, we know that it has equal roots thus its discriminant must equal to zero: d=(-8)^2+4b=0. Solving for b gives b=-16.

How did you get d=(-8)^2+4b=0.?

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propcandy wrote:
Substitute a=-8 in the first equation: x^2-8x-b=0.

Now, we know that it has equal roots thus its discriminant must equal to zero: d=(-8)^2+4b=0. Solving for b gives b=-16.

How did you get d=(-8)^2+4b=0.?


The general form of a quadratic equation is \(ax^2+bx+c=0\). It's roots are:
\(x_1=\frac{-b-\sqrt{b^2-4ac}}{2a}\) and \(x_2=\frac{-b+\sqrt{b^2-4ac}}{2a}\)

Expression \(b^2-4ac\) is called discriminant:
  • If discriminant is positive quadratics has two roots;
  • If discriminant is negative quadratics has no root;
  • If discriminant is zero quadratics has one root.

Check more HERE.
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New post 10 Aug 2015, 16:31
so b^2-4ac becomes (-8)^2+4b? what happens to the negative and the a?

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New post 16 Aug 2015, 06:35
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Please explain how d=(−8)2+4b=0 comes in the question.Kindly explain.

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Re: D01-41 [#permalink]

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New post 03 Oct 2015, 07:11
Bunuel wrote:
Official Solution:

The equation \(x^2 + ax - b = 0\) has equal roots, and one of the roots of the equation \(x^2 + ax + 15 = 0\) is 3. What is the value of b?

A. \(-\frac{1}{64}\)
B. \(-\frac{1}{16}\)
C. \(-15\)
D. \(-16\)
E. \(-64\)


Since one of the roots of the equation \(x^2 + ax + 15 = 0\) is 3, then substituting we'll get: \(3^2+3a+15=0\). Solving for \(a\) gives \(a=-8\).

Substitute \(a=-8\) in the first equation: \(x^2-8x-b=0\).

Now, we know that it has equal roots thus its discriminant must equal to zero: \(d=(-8)^2+4b=0\). Solving for \(b\) gives \(b=-16\).


Answer: D



Dear Bunuel or Engr2012

Can you explain how you plugged in the values: for the expression b^2−4ac which is called discriminant: I understand if it has same roots, then d must equal 0.

But I do not understand the following because you should be plugging in a = -8 into the equation above, ... i can not follow how you then have -8^2 which should be b^2. And what about variable c? Please help.

\(d=(-8)^2+4b=0\). Solving for \(b\) gives \(b=-16\).
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reto wrote:
Bunuel wrote:
Official Solution:

The equation \(x^2 + ax - b = 0\) has equal roots, and one of the roots of the equation \(x^2 + ax + 15 = 0\) is 3. What is the value of b?

A. \(-\frac{1}{64}\)
B. \(-\frac{1}{16}\)
C. \(-15\)
D. \(-16\)
E. \(-64\)


Since one of the roots of the equation \(x^2 + ax + 15 = 0\) is 3, then substituting we'll get: \(3^2+3a+15=0\). Solving for \(a\) gives \(a=-8\).

Substitute \(a=-8\) in the first equation: \(x^2-8x-b=0\).

Now, we know that it has equal roots thus its discriminant must equal to zero: \(d=(-8)^2+4b=0\). Solving for \(b\) gives \(b=-16\).


Answer: D



Dear Bunuel or Engr2012

Can you explain how you plugged in the values: for the expression b^2−4ac which is called discriminant: I understand if it has same roots, then d must equal 0.

But I do not understand the following because you should be plugging in a = -8 into the equation above, ... i can not follow how you then have -8^2 which should be b^2. And what about variable c? Please help.

\(d=(-8)^2+4b=0\). Solving for \(b\) gives \(b=-16\).



Once you get a =-8 and know that as the discriminant for equal roots = 0 ---> from the original equation\(x^2+ax-b=0\),

the discriminant is = \(a^2-4(1)(-b)=0\)

--->\(a^2+4b=0\)--> substitute \(a =-8\) , \((-8)^2+4b=0\) ---> \(64+4b=0\) ---> \(b = -16\).
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You can still get this problem if you're like me and aren't totally comfortable with the concept of a discriminant.

Once you get to the point where you figure out a = -8, rephrase the x^2 + ax - b = 0 equation as x^2 - 8x - b = 0 and simply plug in the answer choices, then factor. The equation has "equal roots" so its going to take the form of (x+y)^2 or (x-y)^2.

Starting with C, the equation x^2 - 8x + 15 = 0, but since 15 isn't a perfect square this clearly won't yield equal roots while adding up to -8x.

Moving on to D, the equation becomes x^2 - 8x + 16 = 0; or (x-4)(x-4)=0. It fits.

The other choices don't fit as they don't add up to -8x when factored/foiled.

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I think this is a high-quality question and I agree with explanation. Great question! But it appears to be more of a 700 range and not 600.
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Re: D01-41 [#permalink]

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New post 21 Feb 2016, 10:29
bdawg2057 wrote:
You can still get this problem if you're like me and aren't totally comfortable with the concept of a discriminant.

Once you get to the point where you figure out a = -8, rephrase the x^2 + ax - b = 0 equation as x^2 - 8x - b = 0 and simply plug in the answer choices, then factor. The equation has "equal roots" so its going to take the form of (x+y)^2 or (x-y)^2.

Starting with C, the equation x^2 - 8x + 15 = 0, but since 15 isn't a perfect square this clearly won't yield equal roots while adding up to -8x.

Moving on to D, the equation becomes x^2 - 8x + 16 = 0; or (x-4)(x-4)=0. It fits.

The other choices don't fit as they don't add up to -8x when factored/foiled.


Thanks, This made my day.

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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. I don't understand what is "we know that it has equal roots thus its discriminant must equal to zero: d=(−8)2+4b=0d=(−8)2+4b=0."means.

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New post 25 May 2016, 08:53
daraepark88 wrote:
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. I don't understand what is "we know that it has equal roots thus its discriminant must equal to zero: d=(−8)2+4b=0d=(−8)2+4b=0."means.


GMAT Club's questions are mostly quite difficulty. One should not attempt them if the fundamentals are not strong enough.

Check below links for some theory on algebra:
Factoring Quadratics: http://www.purplemath.com/modules/factquad.htm
Solving Quadratic Equations: http://www.purplemath.com/modules/solvquad.htm

Theory on Algebra: algebra-101576.html
Algebra - Tips and hints: algebra-tips-and-hints-175003.html

DS Algebra Questions to practice: search.php?search_id=tag&tag_id=29
PS Algebra Questions to practice: search.php?search_id=tag&tag_id=50

Special algebra set: new-algebra-set-149349.html

Hope this helps.
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Re: D01-41   [#permalink] 25 May 2016, 08:53

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