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If a >0 and b <0, which of the following statements are tru
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10 Apr 2017, 13:00
10 Apr 2017, 13:00
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If a >0 and b <0, which of the following statements are true about the values of x that solve the equation \(x^2-ax+b=0\) ?
Indicate all such statements.
A. They have opposite signs.
B. Their sum is greater than Zero.
C. Their product equals -b
Indicate all such statements.
A. They have opposite signs.
B. Their sum is greater than Zero.
C. Their product equals -b
Retired Moderator
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Re: Please Explain this math
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10 Apr 2017, 13:38
10 Apr 2017, 13:38
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monir301 wrote:
If a > 0 and b < 0, which of the following statements are true about the values of x that solve the equation x² - ax + b = 0 ?
Indicate all such statements.
A. They have opposite signs.
B. Their sum is greater than Zero.
C. Their product equals -b
Indicate all such statements.
A. They have opposite signs.
B. Their sum is greater than Zero.
C. Their product equals -b
If we were to FACTOR x^2 - ax + b into the form (x + k)(x + j), we'd have two clues to help us.
First, we know that the product jk must equal b.
Since we're told that b is NEGATIVE, we know that one of the variables (j or k) is POSITIVE, and the other is NEGATIVE.
Second, we know that the sum, j + k, is equal to -a
Since a is positive, we know that -a is NEGATIVE
So, we already know that one of the variables (j or k) is POSITIVE, and the other is NEGATIVE, AND we now know that their SUM is negative.
This means that the negative value (j or k) has a greater magnitude than the positive value. This is the only way that the sum, j + k, can be negative.
So, here are some possible expressions/equations that meet these criteria:
1. x² - 3x - 15 = 0
2. x² - 2x - 8 = 0
3. x² - 5x - 6 = 0
4. x² - 10x - 39 = 0
When we factor and solve them, we get:
1. (x - 5)(x + 3) = 0, so x = 5 or -3
2. (x - 4)(x + 2) = 0, so x = 4 or -2
3. (x - 6)(x + 1) = 0, so x = 6 or -1
4. (x - 13)(x + 3) = 0, so x = 13 or -3
Let's examine the 3 statements:
A. They have opposite signs.
This is true.
B. Their sum is greater than Zero.
This is true.
C. Their product equals -b
This LOOKS true, but it isn't.
For example, in the first equation x² - 3x - 15 = 0, the value of b is -15, which means b = 15
When we solved the equations, we found that x = 5 or -3
So, the product of the solutions is -15 (not 15)
Cheers,
Brent
General Discussion
Re: If a >0 and b <0, which of the following statements are tru
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29 Aug 2022, 09:03
29 Aug 2022, 09:03
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Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
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