Hello
I have equation:Equation has 2 solutions. p and q are integer and positive. How many different p and q exsist if both solutions are less then 5 (x1,x2<5).Code:x^2 - px + q = 0
Sorry for my English. Please help. Cheers.
I do not understand this (x1,x2<5).
Could you explain what that means in this situation?
Since p & q are integers, x must be greater than zero, and an integer.
That's eight solutions.Code:x p q 1 1 0 1 2 1 1 3 2 1 4 3 2 2 0 2 3 2 2 4 4 3 4 3
If you removed x=1, p=1, q=0 and x=2, p=2, q=0, you would still have 6 solutions.
Could you explain the restrictions so that only 2 work?
x1 and x2 are roots
Root (mathematics) - Wikipedia, the free encyclopedia
Sorry for my English
Condition 1:
Since roots are real and distinct:
$\displaystyle Discriminant>0$
$\displaystyle p^2-4q>0$
$\displaystyle 4q<p^2$................................(1)
Condition 2:
Since both roots are less than $\displaystyle 5$,the sum of the roots will be less than $\displaystyle 10$.
$\displaystyle p<10$
Condition 3:
Let $\displaystyle f(x)=x^2-px+q$
$\displaystyle f(5)>0$
$\displaystyle 25-5p+q>0$
From conditions 1,2 and 3 we can see that $\displaystyle 0<p<10$ and $\displaystyle 0<q<25$,where p and q are integers
I hope this helps.
Since $\displaystyle x_{1}$ and $\displaystyle x_{2}$ are roots of $\displaystyle f(x)=0$,so $\displaystyle (x-x_{1}) $ and $\displaystyle (x-x_{2}) $ are factors of f(x)=0
$\displaystyle f(x)=x^2-px+q=(x-x_{1})(x-x_{2})$
Since both the roots are less than $\displaystyle 5$,clearly both $\displaystyle (5-x_{1})$ and $\displaystyle (5-x_{2}) $ are positive.Thus
$\displaystyle f(5)=5^2-p(5)+q=(5-x_{1})(5-x_{2})>0$
Since $\displaystyle 4q<p^2$
$\displaystyle 0<p<10$
$\displaystyle \therefore 4q<p^2<100$
due to which $\displaystyle 0<q<25$
Now these values of $\displaystyle p$ and $\displaystyle q$ must also satisfy the condition $\displaystyle f(5)>0$
$\displaystyle 5p<25+q<25+25$
$\displaystyle p<10$.
Thus the values $\displaystyle 0<p<10$ and $\displaystyle 0<q<25$ also satisfy the condition $\displaystyle f(5)>0$
Thx
But I don't know how many p&q satisfy the condition x^2 - px + q = 0 ?
For exaple in this reason result is 16 (I have written program in c++ and check a lot of p & q, if condition was satisfied r++ and r was a resault).
How can I get that?