M28-08

Question

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

Bunuel · Accepted Answer

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

Since one of the roots of the equation  x^2 + ax + 15 = 0  is 3, then substituting we'll get:  3^3+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: B

JackSparr0w · Answer

I thought the Discriminant was (b^2)-4ac.

This would yeild ((-8)^2)-4(1)(b) --> 64-4x.

So: 64-4x=0 -->64=4x -->x=16.

Can someone explain why this isn't the case? In Bunuel's solution we have addition, instead of subtraction in the discriminant.

Thanks

Bunuel · Answer

Yes, the discriminant of  ax^2 + bx + c = 0  is  b^2 - 4ac .

Thus the discriminant of  x^2 - 8x - b = 0  is :  (-8)^2 - 4(-b) = (-8)^2 + 4b .

Hope it's clear.

JackSparr0w · Answer

Got it, thanks!

filipTGIM · Answer

Isn't it just more simple to acknowledge that x^2-8x-b=0 and then, knowing the roots are same, use reverse foil method find: 4+4=8, 4*4=16. ??

Knowing that Bunuel always uses the simplest method, please explain the importance of knowing the discriminant? I thought with reverse foil the discriminant was not important to remember.

Thanks !

propcandy · Answer

This question was duplicated from Q D01-41. I got this question twice from a quiz I was doing.

HimJ · Answer

Alternative method:

We know one of the roots of x^2 +ax + 15 = 0, so let x=3
a=-8 . . . . . by substituting x=3 in x^2 +ax + 15 = 0

Standard form of quadratic equation is ax^2+bx+c=0, let its root be p and q. so, (x-p)(x-q)=0
Solving (x-p)(x-q)=0, we get x^2 -(p+q)+pq=0

By comparing x^2 -(p+q)+pq=0 with x^2 +ax - b = 0 that is x^2 -8x - b=0 . . . . .as a=-8
we can deduce that p+q=8 and pq=-b
It is given that the roots are equal so both the roots p and q must be 4 as p+q=8
and pq=16

Substituting above values in pq=-b we get that -b=16 so b=-16

If you learned something new, hit kudos button as a gratitude

apurvgodbole · Answer

We know one of the roots of x^2 +ax + 15 = 0 is 3. 
Substitute x=3 in the equation.
We get x= -8 as the other root.
We know that  sum of roots  in equation ax^2 +bx + c = 0,  is -b/a  and  product of the roots is c/a. 
So, in equation x^2 +ax -b = 0, a becomes -8 => equation becomes, x^2 -8x - b = 0
(Let root 1= root 2 = r)
r + r = -8 => r = -4
r*r = -b => b = -16

Ans  B

I was comfortable using the sum and product of roots concept and could solve faster that way. Hence, I thought I might add this as a solution.

Cheers!

Bunuel · Answer

Duplicate of D01-41. Unpublished.

Dumsy_1711 · Answer

Guys I have a doubt.

I tried solving by substituting the values for b. So for b = -16

x^2 -4x-4x -16 = 0

x (x-4) - 4(  x+4  ) =0

But the equation cannot be factored this way. Can anyone tell me where I'm going wrong please?

Bunuel · Answer

When you substitute n = -16 into x^2 - 8x - n = 0 you'd get x^2 - 8x - (-16) = 0, and finally, x^2 - 8x + 16 = 0, not x^2 - 8x - 16 = 0. And x^2 - 8x + 16 = 0 can be written as (x - 4)^2 = 0. Hope it helps. P.S. If still have any questions please continue discussion in this topic: https://gmatclub.com/forum/d01-183502.html THIS TOPIC IS LOCKED.

M28-08

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