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# Inequalities, how to crack solve this

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Inequalities, how to crack solve this [#permalink]  23 Aug 2011, 23:17
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75% (01:34) correct 25% (00:29) wrong based on 1 sessions
If there is exactly one root of the equation x2 + ax + b, where a and b are positive constants, what is b in terms of a?
(A) a/2
(B) a
(C) 3a/2
(D) a2/2
(E) a2/4
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Re: Inequalities, how to crack solve this [#permalink]  23 Aug 2011, 23:48
i guess the source is manhattan...anywayz
i got struck with a simple technique...picking up numbers..
consider (x+1)^2=x^2+2x+1
(x+2)^2=x^2+4x+4
....similarly u can go for other numbers as well..then analyse the values of a and b by considering the given options..
ans e
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Re: Inequalities, how to crack solve this [#permalink]  24 Aug 2011, 00:06
x^2+ax+b => has only 1 solution
So the equation should be of the form x^2 + 2Bx + B^2=0 (i.e) (x+B)^2=0 OR x^2 - 2Bx + B^2=0 (i.e) (x-B)^2=0
Hence substitute in place of B^2 = b and in place of 2B = a
we get a=2\sqrt{b}
=> a^2=4b [Squaring on both sides]
=> b=a^2/4
(E)

Hope this helps !
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Re: Inequalities, how to crack solve this [#permalink]  24 Aug 2011, 00:11
how did you take a=2sqrtb, from the above equation ?
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Re: Inequalities, how to crack solve this [#permalink]  24 Aug 2011, 01:47
naaga wrote:
how did you take a=2sqrtb, from the above equation ?

Since the given equation is a quadratic equation, there would always be 2 roots (solution) for it. eg: (x+p)(x+q)
But the question says the equation has only 1 solution. This could be possible only when both the roots are same. i.e (x+p)(x+p).
This can be written as (x+p)^2
We know the formula for this i.e (x+p)^2 = x^2+2px+p^2
Now visualize the given equation (x^2+ax+b) with the above formula:
Coefficient of x^2 = 1
Coefficient of x = a (which is nothing but 2p)
Constant term (i.e the p^2) = b
Hence we can write:
p^2=b
=> p=\sqrt{b}
=> a=2p = 2\sqrt{b}
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Re: Inequalities, how to crack solve this [#permalink]  24 Aug 2011, 03:36
thank you srivats212, nice explanation.
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Re: Inequalities, how to crack solve this [#permalink]  24 Aug 2011, 12:31
the root must be -a/2. (Quadratic equation with only one root)
Substituting the -a/2 in the equation & equating it to 0.
You will get b= a^2 / 4
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Re: Inequalities, how to crack solve this [#permalink]  24 Aug 2011, 21:02
one root for a quadratic equation ax^2+bx+c is possible only when b^2 = 4ac ---1

Here b = a
c= b
a = 1

substituting these values in 1, we have

a^2 = 4b => b =a^2/4

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Re: Inequalities, how to crack solve this [#permalink]  26 Aug 2011, 06:16
Can be possible only when determinent = 0

therefore, b ^2 - 4ac= 0
b=a , a=1 and c=b. After substituting values we will get a^2 - 4 (1)(b)=0 =>a ^2 =4b =>b= a^2/4.
Re: Inequalities, how to crack solve this   [#permalink] 26 Aug 2011, 06:16
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