Advertisements
Advertisements
Question
Solve the equation x4 + 2x3 - 13x2 + 2x + 1 = 0.
Advertisements
Solution
GIven equation
x4 + 2x3 - 13x2 + 2x + 1 = 0
Dividing both sides by x2, we get
x2 + 2x - 13 +`(2)/x + (1)/x^2` = 0
⇒ `(x^2 + 1/x^2) +2(x + 1/x) -13` = 0
Put `x + (1)/x = y, "squaring" x^2 + (1)/x^2 + 2 = y^2`
⇒ `x^2 + (1)/x^2 = y^2 - 2`
Then y2 - 2 + 2y - 13 = 0
⇒ y2 + 2y - 15 = 0
⇒ y2 + 5y - 3y - 15 = 0
⇒ y(y + 5) -3(y + 5) = 0
⇒ (y + 5)(y - 3) = 0
⇒ y + 5 = 0 or y = -5
or y - 3 = 0 or y =3
But `x + (1)/x = -5`
Then `x + (1)/x = -5`
⇒ x2 + 1 = -5
⇒ x2 + 5x + 1 = 0
⇒ x = `(-b ± sqrt(b^2 - 4ac))/(2a)`
x = `(-5 ± sqrt(25 - 4))/(2 xx 1)`
x = `(-5 ± sqrt(25 - 4))/(2)`
x = `(-5 ± sqrt(21))/(2)`
or `x + (1)/x = 3`
⇒ x2 + 1 = 3
⇒ x2 - 3x + 1 = 0
⇒ x = `(-b ± sqrt(b^2 - 4ac))/(2a)`
x = `(-(-3) ± sqrt(9 - 4))/(2)`
x = `(3 ± sqrt(5))/(2)`
Hence x = `(-5 ± sqrt(21))/(2), (3 ± sqrt(5))/(2)`
RELATED QUESTIONS
The difference of two numbers is 4. If the difference of their reciprocals is 4/21. Find the numbers.
Two natural number differ by 3 and their product is 504. Find the numbers.
A two digit number is such that the product of the digits is 14. When 45 is added to the number, then the digit are reversed. Find the number.
Solve equation using factorisation method:
4(2x – 3)2 – (2x – 3) – 14 = 0
Two pipes flowing together can fill a cistern in 6 minutes. If one pipe takes 5 minutes more than the other to fill the cistern, find the time in which each pipe would fill the cistern.
Solve the following by reducing them to quadratic form:
`sqrt(y + 1) + sqrt(2y - 5) = 3, y ∈ "R".`
Solve the following equation by factorization
x2 – (p + q)x + pq = 0
Solve the following equation by factorisation :
2x2 + ax – a2= 0
The product of two successive integral multiples of 5 is 300. Then the numbers are:
If Zeba were younger by 5 years than what she really is, then the square of her age (in years) would have been 11 more than five times her actual age. What is her age now?
