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805+ (Hard)|   Absolute Values|   Algebra|      
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sarthakaggarwal
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Find K if the number of values of x that satisfy the equation 21 Sep 2021, 08:45
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

95% (hard)

Question Stats:

38% (02:26) correct 62% (02:45) wrong based on 237 sessions

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Find K if the number of values of x that satisfy the equation \(|x^2 - 16x + 55|=K\) is 3

A 7
B 8
C 9
D 10
E 11
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C
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GMAT Focus 1: 735 Q90 V89 DI81
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Find K if the number of values of x that satisfy the equation 21 Sep 2021, 10:51
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sarthakaggarwal
Find K if the number of values of x that satisfy the equation \(|x^2 - 16x + 55|=K\) is 3

A 7
B 8
C 9
D 10
E 11
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\(|x^2 - 16x + 55|=K\)
\(|x^2 - 16x + 64-9|=K\)
\(|(x-8)^2-3^2|=k\)
\(|(x-8-3)(x-8+3)|=k\)
\(|(x-11)(x-5)|=k\)

\(|(x-11)(x-5)|=|(12-11)(12-5)|=|1*7|=7\)
\(|(x-11)(x-5)|=|(4-11)(4-5)|=|-1*-7|=7\)
So x can be 12 or 4

\(|(x-11)(x-5)|=|(7-11)(7-5)|=|-4*2|=8\)
\(|(x-11)(x-5)|=|(9-11)(9-5)|=|-2*4|=8\)
So x can be 7 or 9

\(|(x-11)(x-5)|=|(8-11)(8-5)|=|3*3|=9\)
So x can be 8.


How can k as 9 give 3 values of x??
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revati16
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Find K if the number of values of x that satisfy the equation 30 Sep 2021, 11:10
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Can someone please elaborate on the answer a little. I am unable to understand this question
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the27s
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Find K if the number of values of x that satisfy the equation 30 Sep 2021, 13:54
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Quadratic has 2 solutions (can be distinct or identical), attach a modulus to it, the equation may have 2 more solutions (again distinct or identical).

As it is given that 3 solutions exist for x, one of the 2 quadratics (obtained by expanding the modulus) should have equal real roots.

Testing for b^2-4ac = 0, we get k=9/-9

That allows us to select option C.

But k is supposed to be positive, so this quadratic having 3 solutions seems incorrect.

Maybe the question should've read the equation satisfies 1 value of x.

Posted from my mobile device
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Find K if the number of values of x that satisfy the equation 08 Dec 2021, 01:56
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chetan2u
sarthakaggarwal
Find K if the number of values of x that satisfy the equation \(|x^2 - 16x + 55|=K\) is 3

A 7
B 8
C 9
D 10
E 11


\(|x^2 - 16x + 55|=K\)
\(|x^2 - 16x + 64-9|=K\)
\(|(x-8)^2-3^2|=k\)
\(|(x-8-3)(x-8+3)|=k\)
\(|(x-11)(x-5)|=k\)

\(|(x-11)(x-5)|=|(12-11)(12-5)|=|1*7|=7\)
\(|(x-11)(x-5)|=|(4-11)(4-5)|=|-1*-7|=7\)
So x can be 12 or 4

\(|(x-11)(x-5)|=|(7-11)(7-5)|=|-4*2|=8\)
\(|(x-11)(x-5)|=|(9-11)(9-5)|=|-2*4|=8\)
So x can be 7 or 9

\(|(x-11)(x-5)|=|(8-11)(8-5)|=|3*3|=9\)
So x can be 8.


How can k as 9 give 3 values of x??
Show more

It's not necessary x to be an integer, so for \(k = 9\), \(|x^2 - 16x + 55|=9\) has three solutions:
\(x = 8 - 3 \sqrt {2}\);
\(x = 8 + 3 \sqrt {2}\);
\(x=8\).

If k is 7 or 8, the equation has four solutions, and if k is 10 or 11, the equation has two solutions.
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Find K if the number of values of x that satisfy the equation 29 Sep 2023, 09:45
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Bunuel
chetan2u
sarthakaggarwal
Find K if the number of values of x that satisfy the equation \(|x^2 - 16x + 55|=K\) is 3

A 7
B 8
C 9
D 10
E 11


\(|x^2 - 16x + 55|=K\)
\(|x^2 - 16x + 64-9|=K\)
\(|(x-8)^2-3^2|=k\)
\(|(x-8-3)(x-8+3)|=k\)
\(|(x-11)(x-5)|=k\)

\(|(x-11)(x-5)|=|(12-11)(12-5)|=|1*7|=7\)
\(|(x-11)(x-5)|=|(4-11)(4-5)|=|-1*-7|=7\)
So x can be 12 or 4

\(|(x-11)(x-5)|=|(7-11)(7-5)|=|-4*2|=8\)
\(|(x-11)(x-5)|=|(9-11)(9-5)|=|-2*4|=8\)
So x can be 7 or 9

\(|(x-11)(x-5)|=|(8-11)(8-5)|=|3*3|=9\)
So x can be 8.


How can k as 9 give 3 values of x??

It's not necessary x to be an integer, so for \(k = 9\), \(|x^2 - 16x + 55|=9\) has three solutions:
\(x = 8 - 3 \sqrt {2}\);
\(x = 8 + 3 \sqrt {2}\);
\(x=8\).

If k is 7 or 8, the equation has four solutions, and if k is 10 or 11, the equation has two solutions.
Show more

Please can you explain how K=9?
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Find K if the number of values of x that satisfy the equation Updated on: 15 Dec 2024, 00:29
Originally posted by khanabd8 on 15 Dec 2024, 00:26.
Last edited by khanabd8 on 15 Dec 2024, 00:29, edited 6 times in total.
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sarthakaggarwal
Find K if the number of values of x that satisfy the equation \(|x^2 - 16x + 55|=K\) is 3

A 7
B 8
C 9
D 10
E 11
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Case 1:
x^2 - 16x + 55 - k = 0

Determinant:
(-16)^2 - 4(1)(55 - k)
= 256 - 220 + 4k
= 36 + 4k

Case 2:
-x^2 + 16n - 55 - k = 0

Determinant:
(16)^2 - 4(-1)(-55 - k)
=256 - 220 - 4k=
= 36 - 4k

---

Determinant:
  • (+ve) >0 has 2 solutions
  • = 0 has 1 solution
  • −ve <0 has no real solution (or 2 imaginary solutions not GMAT scope)

- Also: K is always (+ve) >0 as K = absolute Value expression


So for 3 solutions one of the determinants need to be = 0

Case 1:
36+4k always (+ve) >0 as K always (+ve) >0
Case 2:
36-4k =0 (for 1 solution)
-4k = -36
4k = 36
k = 9

Choice: C
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Find K if the number of values of x that satisfy the equation 15 Dec 2024, 01:18
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Find K if the number of values of x that satisfy the equation \(|x^2 - 16x + 55|=K\) is 3

\(|x^2 - 16x + 55| = |(x-11)(x-5)| = K\)

Either x^2-16x+55 = K
x^2 - 16x + (55-K) = 0
Which will have real roots provided 16^2-4(55-K)>=0

Or x^2-16x+55 = -K
x^2 - 16x + (55+K) = 0
Which will have real roots provided 16^2-4(55+K)>=0

If K=0
x^2-16x+55=(x-11)(x-5)=0; x=11 or 5; 2 roots; Not possible since the number of values of x that satisfy the equation is 3

Normally the equation combined will have 4 roots but they can have 3 roots if
1. One of the equation has equal roots
2. The equations have a common root

Case 1. One of the equation has equal roots
a. x^2 - 16x + (55+K) = 0 has equal roots
16^2 - 4(55+K) = 0
K = 9
b. x^2 - 16x + (55-K) = 0 has equal roots
16^2-4(55-K)=0
K = -9

Case 2. The equations have a common root
x will satisfy both equations
x^2 - 16x + (55+K) = 0
x^2 - 16x + (55-K) = 0
K = 0;
x^2-16x+55=(x-11)(x-5)=0; x=11 or 5; 2 roots; Not possible since the number of values of x that satisfy the equation is 3

K = {9,-9}

Since K=-9 is not provided, K = 9

IMO C
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Find K if the number of values of x that satisfy the equation 15 Dec 2024, 12:18
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To get three roots of the two given quadratic is possible when both the roots of an equation are equal.

Here's the detailed solution:

Smart Prep (Tutor)

sarthakaggarwal
Find K if the number of values of x that satisfy the equation \(|x^2 - 16x + 55|=K\) is 3

A 7
B 8
C 9
D 10
E 11
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Find K if the number of values of x that satisfy the equation 17 Dec 2024, 10:55
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one case
x^2 - 16x + 55 = k
second case
x^2 - 16x + 55 = -k

roots for case 1 = (-16+-sqrt{16^2 - 4(55-k)})/2
roots for case 2 = (-16+-sqrt{16^2 - 4(55+k)})/2

To have only 3 roots, one of these under roots must become zero, as 4*64 = 256 which is equal to 16^2
we can make the under root in case 2 as 0, because we can get the value 64 only in the case 2 in (55+k)
16^2 - 4(55+k) = 0
k = 9
sarthakaggarwal
Find K if the number of values of x that satisfy the equation \(|x^2 - 16x + 55|=K\) is 3

A 7
B 8
C 9
D 10
E 11
Show more
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