0
$\begingroup$

IF $a>0$ and $b<0$, which of the following statements are true about the value of (x) that solve the eq0ution $x^2 - ax + b = 0$

a)they have opposite signs b)their sum is greater than zero c)their product equals $- b$

Now my choice was a) and c), but c) is incorrect and I'm not sure why since b is negative (<0) which means that the values of x will be opposite (a negative product) hence a), but why not c)? The answer is a) and b) but I'm not sure why b) is an answer. It COULD be I think, but I believe c) has to be. Thanks everyone.

1 Answers 1

2

$x^2-ax+b=0 \implies $

$x = \frac {a\pm\sqrt {a^2-4b}}2$

$b <0$ so $-4b>0$ so $\sqrt {a^2 -4b} >\sqrt {a^2}= a >0$.

So $a-\sqrt {a^2-4b} < a -a =0$.

So $\frac {a + \sqrt {a^2-4b} }2> 0 >\frac { a -\sqrt {a^2-4b}}2 $

so a)is true.

b) $\frac {a + \sqrt {a^2-4b}}2+ \frac {a - \sqrt {a^2-4b}}2 = a >0$ so b).

And $ \frac {a + \sqrt {a^2-4b}}2 \frac {a -\sqrt {a^2-4b}}2=\frac {a^2-(a^2-4b)}4=b \ne -b $ so not c)

  • 0
    Actually, since the solutions have opposite signs, their product is negative. And as b <0 then -b is positive. So the product can't be -b. You don't have to multiply them out. – 2017-01-29
  • 0
    Ah, I think I see what you're saying. When you look at answer choice c) you're seeing -(b) and since we agree that the product is negative and the product is = b you're seeing -(-b)? I hope this is the case since that's what I was thinking COULD be the case with answer choice c). Basically answer choice c) is saying the answer is POSITIVE? – 2017-01-29
  • 0
    No. I'm not seeing or saying anything of the sort! I am saying $-b \ne b $. It just doesn't. $b $ is a negative number and $-b $ is a positive number. The negative sign in front of a number has *NOTHING* to do with whether the number is negative or positive. $-b=(-1)*b $ and just as likely to be positive as it is to be negative. Remember if $k< 0$ then $|k|= -k$ because absolute values of a negative number is always positive and $-k$ is a positive number. Likewise $|-k|=-k$ because the absolute value of a positive number is a positive number. $-k $ is a POSITIVE number. – 2017-01-29
  • 0
    great thanks for taking the time to help me out. – 2017-02-04