Root Problem

0313phd

New member
Joined
Apr 21, 2011
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Question: If b and c are integers such that the equation 3xsquared + bx + c= 0 has only one real root, which of the following
statements must be true?
1. b is even
2. c is odd
3. bsquared is a multiple of 3.

The problem is I can only guess at this one. I know that if the equation is equal to zero, there is only one real root. That is all I know. #3 seemed plausible if I imagine factoring the polynomial, but the answer includes #1, in addition to #3. I can understand the answer if I imagine factoring the polynomial, but is there a simpler answer to this question that eludes me?
0313
 
CLARIFICATION OF THE PREVIOUS ROOT PROBLEM:
SOMEONE WROTE EARLIER TO ME: "Twice, we've asked you to type polynomials using math notation and the caret symbol instead of English words."

Please type like this:

y = ax^2 + bx + c, where b^2 - 4ac = 0

I just read the above note, so I will retype the polynomial in the question as: 3x> + bx + c
I am using > for the squared notation, instead of the caret symbol that you requested, because I don't see a key for the caret symbol on my keyboard. 0313
 
It looks as if "someone" was also giving you a big hint for your problem solution. For a quadratic equation of the form ax^2 + bx + c = 0 to have only one real root, it must be true that b^2 - 4ac = 0. Therefore,

b^2 = 4ac

In your problem, a = 3, so

b^2 = 3(4c)

BTW, the caret symbol is above the 6 on your keyboard, most likely.
 
0313phd said:
Question: If b and c are integers such that the equation 3xsquared + bx + c= 0 has only one real root, which of the following
statements must be true?
1. b is even
2. c is odd
3. bsquared is a multiple of 3.

The problem is I can only guess at this one. I know that if the equation is equal to zero, there is only one real root. That is all I know. #3 seemed plausible if I imagine factoring the polynomial, but the answer includes #1, in addition to #3. I can understand the answer if I imagine factoring the polynomial, but is there a simpler answer to this question that eludes me?
0313

WHOA

First of all an equation of the form ax^2 + bx + c = 0 can have 2 real roots, 1 real root, or 0 real roots.

It will have 2 real roots if (b^2 - 4ac) > 0.
It will have 1 real root if (b^2 - 4ac) = 0.
It will have no real roots if (b^2 - 4ac) < 0.

(b^2 - 4ac) is called the discriminant of the quadratic.

So in your problem you know that b^2 - 4ac = 0 because there is only one root.
And you know that a = 3 and b and c are integers.
So, b^2 = 12c.

SO ....
 
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