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If x^2+200x+9991=0, which of the following is a value of x?

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If x^2+200x+9991=0, which of the following is a value of x?  [#permalink]

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New post 30 Sep 2018, 17:29
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if \(x^2+200x+9991=0\), which of the following is a value of x?

A. -87

B. -93

C. -101

D. -103

E. -107

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If x^2+200x+9991=0, which of the following is a value of x?  [#permalink]

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New post 30 Sep 2018, 22:23
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1
PeepalTree wrote:
if \(x^2+200x+9991=0\), which of the following is a value of x?

A. -87

B. -93

C. -101

D. -103

E. -107


The question can be done in following ways..
1) substitution -
substitute the choices and see what fits in
2) solve for x
\(x^2+200x+9991=0...........x^2+200x=-9991........x(x+200)=-9991\)

Now it is all about finding factors.... But a bit of logic can help us solve it WAY faster

9991 is NOT a multiple of 3, so BOTH x and x+200 should not be a multiple of 3..

let us see

A. -87.....Multiple of 3....eliminate

B. -93....Multiple of 3....eliminate

C. -101......200-101=99....Multiple of 3....eliminate

D. -103......200-103=97 and 103 both are not multiple of 3, so OK

E. -107.........200-107=93.......Multiple of 3....eliminate

D
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Re: If x^2+200x+9991=0, which of the following is a value of x?  [#permalink]

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New post 30 Sep 2018, 18:16
The best way to solve is to see 9991 is divisible by which of the following...

87,93 divisible by 3.....ruled out

rest is brute force
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Re: If x^2+200x+9991=0, which of the following is a value of x?  [#permalink]

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New post 30 Sep 2018, 18:35
Prime factorization of 9991 is 97 and 103

Solving Quadratic equation

(x+97)(x+103)

Ans x=-103


Ans :

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Re: If x^2+200x+9991=0, which of the following is a value of x?  [#permalink]

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New post 30 Sep 2018, 18:51
PeepalTree wrote:
if \(x^2+200x+9991=0\), which of the following is a value of x?

A. -87

B. -93

C. -101

D. -103

E. -107


Here is what I did...

\(x^2+200x+9991=0\)
\(x^2 + 200x = -9991\)

Now, sum of the units digit of \(x^2\) and \(200x\) must equal \(1\). We'll start from (C).

(C) \((-101)^2 + 200(-101) = (100+1)^2 - (200*101) = 10201 - 20200\). Units digit equals 9, so No.

(D) \((-103)^2 + 200(-103) = (100+3)^2 - (200*103) = 10609 - 20600 = -9991\). That's the answer.
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Re: If x^2+200x+9991=0, which of the following is a value of x?  [#permalink]

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New post 30 Sep 2018, 21:17
PeepalTree wrote:
if \(x^2+200x+9991=0\), which of the following is a value of x?

A. -87

B. -93

C. -101

D. -103

E. -107


The crux of solving this question is figuring out how to quickly factorise the equation by splitting the middle term. The numbers are not easy ones such as 5x or 4x.

But there is enough clue in the numbers.

We have to split 200 into two parts, say p and q. p + q = 200 and pq = 9991
The product pq will be maximum when p = q = 100. At that point pq = 10,000 which is more than 9991.
If we add a k to 100 to get p and subtract a k from 100 to get q, the sum will remain 200.

If we do that, we have to find (100 + k)(100 - k) = 9991.
i.e., 10,000 - k^2 = 9991 or k = 3.
So, the value of p and q are 103 and 97. Or the middle term is split as 103 and 97.

Now that we know that 200 splits as 103 and 97, the answer is (x + 103)(x + 97) = 0
So, values of x are -103 and -97
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If x^2+200x+9991=0, which of the following is a value of x?  [#permalink]

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New post Updated on: 01 Oct 2018, 02:32
Roots of the quadratic equations are\(\frac{-b + \sqrt{b^2 - 4ac}}{2a}\) & \(\frac{-b - \sqrt{b^2 - 4ac}}{2a}\)


From the equation, a = 1, b = 200 c = 9991
So the roots are -97 and -103


Answer: D

Originally posted by magbot on 30 Sep 2018, 21:41.
Last edited by magbot on 01 Oct 2018, 02:32, edited 1 time in total.
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Re: If x^2+200x+9991=0, which of the following is a value of x?  [#permalink]

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New post 22 Oct 2018, 09:41
Veritas Prep Official Solution:

Explanation: In difficult algebra questions, recognition of the common algebraic equations is often the key to success.

In this problem, you must see that \(x^2+200x+9991=0\) is very close to the perfect square \(x^2+200x+10,000\) which is the same as \((x+100)^2\).

Given that, it must be true that \(x^2+200x+9991=0\) is the same as \((x+100)^2–9=0\) or \((x+100)^2=9\).

By inspection you know that x + 100 must equal 3 or -3 so x must be -97 or -103.
Answer is (D).

This skill is called completing the square and can be used anytime you are given a quadratic equation that is close to a perfect square.
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Re: If x^2+200x+9991=0, which of the following is a value of x?  [#permalink]

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New post 22 Oct 2018, 11:14
PeepalTree wrote:
if \(x^2+200x+9991=0\), which of the following is a value of x?

A. -87

B. -93

C. -101

D. -103

E. -107



we can write:
x^2 + 200x + 10,000 = 9
(x+100)^2 = 9

only D satisfies this equation.

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Re: If x^2+200x+9991=0, which of the following is a value of x?  [#permalink]

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New post 23 Oct 2018, 18:23
PeepalTree wrote:
if \(x^2+200x+9991=0\), which of the following is a value of x?

A. -87

B. -93

C. -101

D. -103

E. -107


Notice that 9991 = 10,000 - 9. Because both 10,000 and 9 are perfect squares, we can re-express the difference of squares (10,000 - 9) as (100 - 3)(100 + 3) = (97)(103). So we have:

x^2 + 200x + 9991 = 0

(x + 97)(x + 103) = 0

x = -97 or x = -103

Answer: D
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Re: If x^2+200x+9991=0, which of the following is a value of x? &nbs [#permalink] 23 Oct 2018, 18:23
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