If p≠q and p² = 5p-3 and q² = 5q-3 the equation having roots as p/q and q/p is
well
p = (5±√13)/2
q = (5±√13)/2
If p≠q then one is (5+√13)/2 and the other is (5-√13)/2
(5+√13)/(5-√13) = (19+5√13)/6
(5-√13)/(5+√13) = (19-5√13)/6
So, the equation is
(x-((19+5√13)/6))(x-((19-5√13)/6)) = 0
(6x-19-5√13)(6x-19+5√13) = 0
(6x-19)^2 - 325 = 0
36x^2 - 228x + 36 = 0
3x^2 - 19x + 3 = 0
Thanks dear👍
To find the equation with roots p/q and q/p, we can make use of Vieta's formulas.
Vieta's formulas state that if a quadratic polynomial is given by the equation ax^2 + bx + c = 0, then the sum of the roots is -b/a and the product of the roots is c/a.
Let's start by finding the sum and product of the roots p and q using the given equations:
Given: p² = 5p - 3 ...(1)
q² = 5q - 3 ...(2)
We'll start with equation (1):
p² = 5p - 3
Rearranging this equation, we have:
p² - 5p + 3 = 0
Comparing this equation with ax² + bx + c = 0, we can see that a = 1, b = -5, and c = 3.
By Vieta's formulas, the sum of the roots p and q is given by:
p + q = -b/a = -(-5)/1 = 5 ...(3)
Now let's consider equation (2):
q² = 5q - 3
Rearranging this equation, we have:
q² - 5q + 3 = 0
Again, comparing this equation with ax² + bx + c = 0, we can see that a = 1, b = -5, and c = 3.
By Vieta's formulas, the product of the roots p and q is given by:
pq = c/a = 3/1 = 3 ...(4)
Now we have the sum of the roots (p + q = 5) and the product of the roots (pq = 3).
The equation with roots p/q and q/p can be written as:
x² - (p/q + q/p)x + 1 = 0
We can substitute the values of p + q = 5 and pq = 3 into this equation:
x² - (5)x + 3 = 0
So, the equation with roots p/q and q/p is:
x² - 5x + 3 = 0.
To find the equation with roots p/q and q/p, we can start by finding the sum and product of these roots. Let's assign a variable x to each root, such that x = p/q and y = q/p.
Using the property of reciprocals, we have the following relationships:
xy = 1 (since p/q * q/p = 1)
x + y = (p/q) + (q/p) = (p² + q²)/(pq)
Now, let's find the values of x + y and xy by substituting the given equations for p² and q²:
x + y = (p² + q²)/(pq) = (5p - 3 + 5q - 3)/(pq) = (5p + 5q - 6)/(pq)
xy = 1
Now, we can construct the equation with roots x and y using the sum and product of roots:
Since the equation with roots x and y is of the form (t - x)(t - y) = 0, where t is a variable, the equation can be written as:
(t - x)(t - y) = 0
Expand this equation:
t² - (x + y)t + xy = 0
Substitute the values of x + y and xy we found earlier:
t² - (5p + 5q - 6)/(pq) * t + 1 = 0
This equation represents the equation with roots p/q and q/p.