A community for students.
Here's the question you clicked on:
 0 viewing
mathmath333
 one year ago
What does the following statement means ?
mathmath333
 one year ago
What does the following statement means ?

This Question is Closed

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0\(\large \color{black}{\begin{align} &\normalsize \text{If}\ a=1,\ b,c\in I\ \text{and the roots are rational numbers.}\hspace{.33em}\\~\\ &\normalsize \text{then the root must be an integer} \hspace{.33em}\\~\\ \end{align}}\)

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2Oh, this. Look at the quadratic formula.\[x = \frac{b \pm \sqrt{b^2  4ac}}{2a}\]If \(a=1\):\[x = \frac{b \pm \sqrt{b^2  4c}}{2}\]Now it says that the roots are rational, so basically \(b^2  4c\) is a perfect square. Now it's given that \(b,c\) are integers. If \(b\) is even, then \(b = 2k\) so the root must be numerator must be an even number (why?). Otherwise, if \(b\) is odd then \(b = 2k + 1\) so the root is odd, which is then added to an odd number and made even again. This makes the numerator even, and so the root of the quadratic equation is an integer in both cases.

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0how does \(b=2k\pm1\) makes the other root integer ?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2Sorry, I hurried through the reply. Let me restate that. It says that the roots are rational, so \(b^2  4c\) is a perfect square. Look at the numerator, i.e., \(b \pm \sqrt{b^2  4c}\). If \(b= 2k \) then \(b^2  4c\) is obviously even, so its square root, which is an integer, is also even. The whole numerator is also even in that case. You can divide that by 2. Similarly for odd \(b = 2k +1\), so then \(b^2  4c\) is odd here, so its square root is odd. But \(b \) is odd too, so you are adding an odd number to an odd number, making it even. So you can divide that by 2.

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0i have one more statement

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0\(\large \color{black}{\begin{align} &\normalsize \text{if a quadratic equation in }\ x\ \text{has more than two roots} \hspace{.33em}\\~\\ &\normalsize \text{then it is an identity in }\ x. \hspace{.33em}\\~\\ \end{align}}\)

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2Yeah, of course! The Fundamental Theorem of Algebra says that the number of complex roots of a polynomial equation is always equal to its degree. If you see a quadratic equation where three values of \(x\) satisfy the equation, then it will, in fact, work for any value of \(x\) (which is what an identity is).

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2I remember this example:\[\dfrac{(xa)(xb)}{(ca)(cb)} + \frac{(xb)(xc)}{(ab)(ac)}+ \frac{(xc)(xa)}{(bc)(ba)}=1\]If you plug in \(a,b,c\), you can see that all of the three satisfy this equation. Without any further trouble, you can claim that this equation is an identity.

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0what does "an identity" means here

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2An identity is something that is true for all values of the variable.\[\sin^2 x + \cos^2 x = 1\]The above is not an equation. It is an identity. It's true for all values of \(x\). You don't need to solve it.

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0are there any other examples of quadratic equation having more than 2 roots.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2Let me rewrite that... If a quadratic equation has more than two roots, then it has infinitely many roots.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2\[\frac{x^2 +x}{x} = x + 1\] is a "quadratic equation" with infinitely many roots.
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.