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Nature of roots of quadratic equations

by thornluke
Tags: equations, nature, quadratic, roots
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thornluke
#1
Dec10-11, 01:18 AM
P: 37
1. The problem statement, all variables and given/known data
The equation kx2 - 3x + (k+2) = 0 has two distinct real roots. Find the set of possible values of k.


2. Relevant equations
Since the equation has two distinct real roots, b2 - 4ac > 0

3. The attempt at a solution
b2-4ac>0
9-4(k+2)(k)>0
9-4(k2+2k) >0
9-4k2-8k>0
= -4k2-8k+9>0
Multiply both sides by -1,
4k2+8k-9>0
(4k-3)(k+3)>0

-3<k<3/4

However the answer is -2.46<k<0.458
I'm lost, help please!
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SammyS
#2
Dec10-11, 01:48 AM
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Quote Quote by thornluke View Post
1. The problem statement, all variables and given/known data
The equation kx2 - 3x + (k+2) = 0 has two distinct real roots. Find the set of possible values of k.


2. Relevant equations
Since the equation has two distinct real roots, b2 - 4ac > 0

3. The attempt at a solution
b2-4ac>0
9-4(k+2)(k)>0
9-4(k2+2k) >0
9-4k2-8k>0
= -4k2-8k+9>0
Multiply both sides by -1,
4k2+8k-9>0
(4k-3)(k+3)>0

-3<k<3/4

However the answer is -2.46<k<0.458
I'm lost, help please!
Your factoring is incorrect.

(4k-3)(k+3) = 4k2 + 9k - 9 .

Also, multiplying by -1 will change > to < .

Solve 4k2+8k-9 = 0 by using the quadratic formula --- or by completing the square.
thornluke
#3
Dec10-11, 02:03 AM
P: 37
Quote Quote by SammyS View Post
Your factoring is incorrect.

(4k-3)(k+3) = 4k2 + 9k - 9 .

Also, multiplying by -1 will change > to < .

Solve 4k2+8k-9 = 0 by using the quadratic formula --- or by completing the square.
((-8 √208)/8) < 0

-2.80 < k < 0.80

I'm getting closer to the "answer" (-2.46<k<0.458) am I wrong, or is the "answer" wrong?

eumyang
#4
Dec10-11, 06:11 AM
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P: 1,347
Nature of roots of quadratic equations

I'm getting the same roots (-2.80 and 0.80).

When I played around with the coefficients of the original quadratic, I found that if you made the coefficient of the x2 term 2k:
2kx2 - 3x + (k+2) = 0
You will get the original answer that you stated: -2.4577 < k < 0.4577. So it looks like either you copied the problem incorrectly or the book has a typo somewhere.
SammyS
#5
Dec10-11, 02:32 PM
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The text book's answer is consistent with -8k2 - 16k +9 > 0 . equivalent to -4k2 - 8k + 9/2 >0

It's hard to see how that's from a simple Typo -- unless the coefficient of x is should have been 3/√2 in the initial equation.
thornluke
#6
Dec12-11, 04:36 AM
P: 37
Quote Quote by eumyang View Post
I'm getting the same roots (-2.80 and 0.80).

When I played around with the coefficients of the original quadratic, I found that if you made the coefficient of the x2 term 2k:
2kx2 - 3x + (k+2) = 0
You will get the original answer that you stated: -2.4577 < k < 0.4577. So it looks like either you copied the problem incorrectly or the book has a typo somewhere.
I guess everyone makes mistakes.. it gets extremely annoying when textbooks provide you with the wrong answers or have a typo.
epenguin
#7
Dec13-11, 05:12 PM
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Quote Quote by thornluke View Post
I guess everyone makes mistakes.. it gets extremely annoying when textbooks provide you with the wrong answers or have a typo.
They are put there to help you to build self-confidence.


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