Find the roots of the equation 2x ² - 9x - 18 = 0.

(x - 1) (2x - 5) = 0 ? x = 1, 5/2
So, its roots are real.

Clearly, 7x² = 49 or 7x² - 49 = 0, which is of the form ax² + bx + c = 0, where b = 0.
Thus, 7x² - 49 = 0 is a quadratic equation.

let us assume the ten's digit number be x and unit's place digit be y.
Then two digit Original Number = 10x + y
According to question ? y = 2x - 1 ................(1)
When digits are interchanged then new number = 10y + x
If the digits at unit's and ten's place are interchanged, the difference between the new and the original number is less than the original number by 20,
New Number - Original Number = Original number - 20
? 10y + x - (10x + y) = 10x + y - 20
? 10y + x - 10x - y - 10x - y = - 20
? 8y - 19x = - 20
? 19x - 8y = 20 ...................(2)
Put the value of y from equation (1) in above equation (2).
? 19x - 8(2x - 1) = 20
? 19x - 16x - 8 = 20
? 7x = 20 + 8
? 7x = 28
? x = 28/7
? x = 4
Put the value of y in equation (1)
From (i) y = 2 x 4 -1 = y = 7
? original number = 10x + y =10 x 4 + 7 = 47.

Let total bill would be ? P
Each one will to pay = ? P/11
10 friends could pay 10 x 60 = ? 600
According to question
600 + P/11 + 50 = P
? 650 = P - P/11
? 650 = (11P - P)/11
? 650 x 11 = (11P - P)
? 650 x 11 = 10P
? P = 11 x 65
? P = 715
Amount paid by 11^th friend = 715/11 + 50 = ? 115

Let us assume the number of brown socks = B
Let us assume the price of brown socks = ? P per pair
Then price of black socks = ? 2P per pair
Before interchanging the shocks the amount = amount of black shocks + amount of brown shocks
? Before interchanging the shocks the total amount = 4 x 2P + B x P
? Before interchanging the shocks the total amount = 8P + BP
After interchanging the shocks the total amount = amount of black shocks + amount of brown shocks
? After interchanging the shocks the total amount = 4 x P + B x 2P
? After interchanging the shocks the total amount = 4P + 2BP
According to question,
After interchanging the shocks the total amount = Before interchanging the shocks the amount + Before interchanging the shocks the amount x 50 %
4P + 2BP = 8P + BP + (8P + BP ) x 50%
? 4P + 2BP = 8P + BP + (8P + BP ) x 50/100
? 4P + 2BP = 8P + BP + (8P + BP ) x 1/2
? 4P + 2BP = ( 16P + 2BP + 8P + BP ) x 1/2
? (4P + 2BP) x 2 = ( 16P + 2BP + 8P + BP )
? 8P + 4BP = 24P + 3BP
? 8P + 4BP = 24P + 3BP
? 4BP - 3BP = 24P - 8P
? BP = 16P
? B = 16
The ratio of the number of black and brown pairs of socks = Number of block shocks / Number of Brown Shocks
? The ratio of the number of black and brown pairs of socks in the original order = 4/16 = 1/4 = 1: 4

Given, one root of x² - 6kx + 5 = 0 is 5.
? x = 5 satisfies x² - 6kx + 5 = 0
? 5² - 30k + 5 = 0
? 25 - 30k + 5 = 0
? 30 - 30k = 0
? 30k = 30
? k = 1

2x² -11x + 15 = 0
[by factorisation method]
? 2x² - (6x + 5x) + 15 = 0
? 2x² - 6x - 5x + 15 = 0
? 2x(x - 3) - 5 (x - 3) = 0
? (2x - 5) (x - 3) = 0
? x = 5/2, 3
Hence, the roots are 5/2 and 3.

[a + (1/a)]² = 3
Taking square roots both sides, we get
a + (1/a) = ?3
On cubing both sides, we
[a + (1/a)]³ = (?3)³
? a³ + 1/a³ + 3.a.1/[a(a + 1/a)] = 3 ?3
? a³ + 1/a³ + 3?3 = 3?3
? a³ + 1/a³ = 0

2x² - 7xy + 3y² = 0
? 2x² - 6xy - xy + 3y² = 0
? 2x(x - 3y) - y(x - 3y) = 0
? (2x - y) (x - 3y) = 0
Either, 2x - y = 0 ? 2x = y
? x/y = 1/2
or x - 3y = 0
? x = 3y ? x/y = 3/1

Given quadric equation is
x² + x/b + 1/c = 0 ...(i)
Now, by condition the roots of the Eq.(i) are ? and 1/?.
Now,product of roots = (1/c) / (1/a)
? ?.(1/?) = a/c
? c = a
which is the required relation.