Suppose a and b are the roots of the quadratic equation x^2 +3x +5=0.
What is the monic quadratic equation with roots a + 1/a and b +1/b?
To find the monic quadratic equation with roots a + 1/a and b + 1/b, we can start by finding the value of (a + 1/a) and (b + 1/b).
Given the roots of the quadratic equation x^2 + 3x + 5 = 0, we know that the sum of the roots is equal to -b/a and the product of the roots is equal to c/a, where the equation is in the form ax^2 + bx + c = 0.
In this case, a = 1, b = 3, and c = 5. So, the sum of the roots is -3/1 = -3 and the product of the roots is 5/1 = 5.
Now, let's find the value of (a + 1/a):
a + 1/a = (a^2 + 1)/a
Since a is one of the roots of the quadratic equation x^2 + 3x + 5 = 0, we can substitute the value of a into the equation:
a^2 + 3a + 5 = 0
Moving the terms around, we get:
a^2 + 5 = -3a
Adding 1 to both sides:
a^2 + 6 = -3a + 1
Now, we can rewrite (a + 1/a):
(a + 1/a) = (a^2 + 6)/(a)
Similarly, we can find the value of (b + 1/b):
(b + 1/b) = (b^2 + 6)/(b)
Now, to find the monic quadratic equation with roots (a + 1/a) and (b + 1/b), we need to express it in the form of ax^2 + bx + c = 0.
Since the equation is monic (the coefficient of x^2 is 1), we can start by multiplying (a + 1/a) and (b + 1/b) together:
(a + 1/a) * (b + 1/b) = ((a^2 + 6)/(a)) * ((b^2 + 6)/(b))
Expanding the expression, we get:
(a^2 * b^2 + 6a^2 + 6b^2 + 36)/(ab)
Multiplying both sides by ab to eliminate the denominator:
ab * (a + 1/a) * (b + 1/b) = (a^2 * b^2 + 6a^2 + 6b^2 + 36)
Rearranging the terms, we have:
a^2 * b^2 + 6a^2 + 6b^2 + 36 - ab * (a + 1/a) * (b + 1/b) = 0
This is the monic quadratic equation with roots (a + 1/a) and (b + 1/b).
Just use your sum and product of roots of a quadratic property.
Here a+b = -3 and ab = 5
sum of new roots = a + 1/a + b +1/b
= a+b + 1/a + 1/b
= a+b + (b+a)/ab = -3 + -3/5 = -18/5
product of new roots = (a + 1/a)(b +1/b)
= ab + a/b + b/a + 1/ab
= 5 + (a^2 + b^2)/ab + 1/ab
= 5 + (a^2 + b^2)/5 + 1/5
So what is a^2 + b^2 ???
well, (a+b)^2 = a^2 + 2ab + b^2
so a^2 + b^2 = (a+b)^2 - 2ab = 9 - 10 = -1
so back to
5 + (a^2 + b^2)/5 + 1/5
= 5 + -1/5 + 1/5 = 5
your new equation is
x^2 + 18/5 x + 5 = 0
or 5x^2 + 18x + 25 = 0