- #1
Asla
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Homework Statement
The total number of integers that satisfy the equation x2-4xy+5y2+2y-4=0
Yea I got that but I still do not know how to continue.Forgive my ignorancelurflurf said:The bound is obvious after the squares are completed. There are a few forms the most useful is
(x+ay)^2+(y+b)^2+c=0
expanding
x^2+2axy+(a^2+1)y^2+2by+(c+b^2)=0
matching with
x^2-4xy+5y^2+2y-4=0
gives
2a=-4
a^2+1=5
2b=2
c+b^2=-4
Let me see,..why do you say (y+b)^2<=-clurflurf said:so you know a,b,c
(x+ay)^2+(y+b)^2+c=0
c<0
(y+b)^2<-c
-b-sqrt(-c)<=y<=-b+sqrt(-c)
is the desired bound on y
list out possibilities
y=-3,-2,-1,0,1,2
since (x+ay)^2+(y+b)^2+c=0
what values of a^2+b^2=5
are lattice points (integers)?
let
a=(x+ay)
b=(y+b)
complete solution
Wow nice,...I guess I will take some time to go over the whole equation again.Thanks for the assistancelurflurf said:(x+ay)^2+(y+b)^2+c=0
(y+b)^2=-(x+ay)^2-c
0<=(x+ay)^2
(y+b)^2=-c
if that is a bit obtuse consider
1+4=5
notice 1,4,5=>0
see that we can conclude
1<5
in general if
a+b=c
a,b,c=>0
b<=c
A quadratic equation is a polynomial equation of the second degree, meaning it has a variable raised to the power of 2. It is written in the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.
Integer solutions are values of the variable in a quadratic equation that result in a whole number when plugged into the equation. In other words, these are values that make the equation true and do not include any fractions or decimals.
To solve a quadratic equation for integer solutions, you can use the quadratic formula or factorization method. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the constants from the equation. The factorization method involves finding two numbers that multiply to equal the constant term (c), and also add up to equal the coefficient of the linear term (b). These two numbers can then be used to rewrite the equation in the form (x + p)(x + q) = 0, where p and q are the two numbers found.
Yes, a quadratic equation can have two integer solutions. This occurs when the discriminant (b^2 - 4ac) is a perfect square, meaning the square root can be simplified to a whole number. For example, the equation x^2 - 4x + 3 = 0 has two integer solutions, x = 1 and x = 3.
If a quadratic equation does not have any integer solutions, it means that the discriminant is not a perfect square and the square root cannot be simplified to a whole number. In this case, the solutions would be irrational numbers or imaginary numbers. For example, the equation x^2 + 2 = 0 does not have any integer solutions, as the square root of -8 is an imaginary number.