If ? and ? are the roots of the equation 4x ² - 19x + 12 = 0, find the equation having the roots 1/? and 1/?

Given that, the roots of the quadrictic equation are 3 and -1.
Let ? = 3 and ? = -1
Sum of roots = ? + ? = 3 - 1 = 2
Products of roots = ? . ? = (3) (-1) = -3
? Required quadric equation is
x² - (? + ?)x + ? ? = 0
? x² - (2)x + (-3) = 0
? x² - 2x - 3 = 0

Given equation is
x² + 2(k - 4)x + 2k = 0
On comparing with ax² + bx + c = 0
Here, a = 1, b = 2(k -4), c = 2k
Since, the root are equal, we have D = 0.
b² - 4ac = 0
? 4(k - 4)² - 8k = 0
4(k² + 16 - 8k) - 8k = 0
? 4k² + 64 - 32k - 8k = 0
? 4k² - 40k + 64 = 0
? k² - 10k + 16 = 0
? k² - 8k - 2k + 16 = 0
? k(k - 8) -2 (k - 8) = 0
? (k - 8) (k - 2) = 0
Hence, the value of k 8 or 2.

Given quadratic equation is
7y² - 50y + k = 0
If one root is 7, then it will satisfy the equation i.e putting y = 7 in equation
7 x (7)² - 50 x 7 + k = 0
? 7 x 49 - 350 + k = 0
? 343 - 350 + k = 0
? k = 7

Given equation is
x² - 3x + 1 = 0 ? x² + 1 = 3x
? x² + 1/x = 3
? x²/x + 1/x = 3
? x + 1/x = 3

Let ? and ? be the roots of the quadratic equation 2x² - 11x + 5 = 0
? ? + ? = -(-11)/2 = 11/2 ...(i)
and ?? = 5/2
Now, (? - ?)² = (? + ?)² - 4??
= (11/2)² - 4(5/2) = 121/4 - 20/2
= 121 - 40/4 = 81/4 = (9/2)²
? Difference of roots = (? - ?) = 9/2 = 4.5

Let ? = 2, ? = 1/4
Then, ? + ? = 2 + 1/4 = 9/4
?? = 2.1/4 = 1/2
Equation having the roots ? and ? is
x² - (? + ?)x + ? ? = 0
? x² - 9/4x + 1/2 = 0
? 4x² - 9x + 2 = 0

Since, p and q are the roots of the equation x² + px + q = 0
Then, p + q = - p
and pq = q
Now, pq = q
? p = 1
Putting the value of p in p + q = - p, we get
1 + q = - 1
? q = - 2

We know that, AM ? GM
? ?a + 1/?a ? 2
Here, ?x² - x + 1 + 1/?x² - x + 1 ? 2
? 2 - x² ? 2
? x² ? 0
? x = 0
Hence, the given equation has only one solution.

Let the roots be ? and ? .
Then, ? + ? = 8 ......(i)
? - ? = 4 ..(ii)
On solving Eqs. (i) and (ii), we get
? = 6, ? = 2
? Required equation is
x² - (? + ?) x + (? ?) = 0
? x² - (6 + 2)x + 6 x 2 = 0
? x² - 8x + 12 = 0