Advertisements
Advertisements
Question
Solve the equation:
`6(x^2 + (1)/x^2) -25 (x - 1/x) + 12 = 0`.
Advertisements
Solution
Given equation
`6(x^2 + (1)/x^2) -25 (x - 1/x) + 12 = 0`
Put x `-(1)/x = y, "squaring" (x - 1/x)^2 = y^2`
⇒ `x^2 + (1)/x^2 - 2 = y^2`
⇒ `x^2 + (1)/x^2 = y^2 + 2`
Now, given equation becomes
6(y2 + 2) - 25y + 12 = 0
⇒ 6y2 + 12 - 25 + 12 = 0
⇒ 6y2 - 25y + 24 = 0
⇒ 6y2 - 16y - 9y + 24 = 0
⇒ 2y(3y - 8) - 3(3y - 8) = 0
⇒ (3y - 8) (2y - 3) = 0
⇒ 3y - 8 = 0 or 2y - 3 = 0
⇒ 3y = 8 or 2y = 3
⇒ y = `(8)/(3)` or y = `(3)/(2)`
But `x - (1)/x = y`
∴ `x - (1)/x = (8)/(3)`
⇒ `(x^2 - 1)/x = (8)/(3)`
⇒ 3x2 - 3 = 8x
⇒ 3x2 - 8x - 3 = 0
⇒ 3x2 - 9x + x - 3 = 0
⇒ 3x(x - 3) + 1(x - 3) = 0
⇒ (x - 3) (3x + 1) = 0
⇒ x - 3 = 0 or 3x + 1 = 0
⇒ x = 3 or x = `(-1)/(3)`
or
`x - (1)/x = (3)/(2)`
⇒ `(x^2 - 1)/x = (3)/(2)`
⇒ 2x2 - 2 = 3x
⇒ 2x2 - 3x - 2 = 0
⇒ 2x2 - 4x + x - 2 = 0
⇒ 2x(x - 2) + 1(x - 2) = 0
⇒ (x - 2) (2x + 1) = 0
⇒ x - 2 = 0 or 2x + 1 = 0
⇒ x = 2 or x = `(-1)/(2)`
Hence, x = 3, `(-1)/(3), 2 and (-1)/(2)`.
RELATED QUESTIONS
An aeroplane take 1 hour less for a journey of 1200 km if its speed is increased by 100 km/hr from its usual speed. Find its usual speed.
Solve each of the following equations by factorization:
`X^2 – 10x – 24 = 0 `
The sum of the squares of two consecutive positive integers is 365. Find the integers.
Find the value of p for which the quadratic equation
\[\left( p + 1 \right) x^2 - 6(p + 1)x + 3(p + 9) = 0, p \neq - 1\] has equal roots. Hence, find the roots of the equation.
Disclaimer: There is a misprinting in the given question. In the question 'q' is printed instead of 9.
If \[x = - \frac{1}{2}\],is a solution of the quadratic equation \[3 x^2 + 2kx - 3 = 0\] ,find the value of k.
Solve the following equation: 4x2 - 13x - 12 = 0
The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages was 124. Determine their present ages.
Solve equation using factorisation method:
2(x2 – 6) = 3(x – 4)
In an auditorium, the number of rows are equal to the number of seats in each row.If the number of rows is doubled and number of seats in each row is reduced by 5, then the total number of seats is increased by 375. How many rows were there?
A piece of cloth costs Rs. 300. If the piece was 5 metres longer and each metre of cloth costs Rs. 2 less, the cost of the piece would have remained unchanged. How long is the original piece of cloth and what is the rate per metre?
