If the roots of the equation x^3 -12x^2 +39x -28 =0 are in ap then their common difference will be?
Answers
Given : Roots of the equation x³ - 12x² + 39x - 28 = 0 are in AP
To find : Common difference of the AP
Solution :
Let's assume that a and d be the first term and common difference of the given AP respectively.
Given the quadratic equation
- x³ - 12x² + 39x - 28 = 0
Here
- a = Coefficient of x³ = 1
- b = Coefficient of x² = -12
- c = Coefficient of x = 39
- d = Constant term = - 28
Now let's assume that x, y and z be the roots of the given equation.
Therefore
- x = a - d
- y = a
- z = a + d
[ This is the method for selecting 3 terms of an AP for the easy calculations ]
Sum of zeroes of an equation is given by
- Sum of zeroes = - b / a
Therefore
⇒ x + y + z = - b / a
⇒ a - d + a + a + d = -(-12) / 1
⇒ 3a = 12
⇒ a = 12 / 3
⇒ a = 4
Product of the zeroes of an equation is given by
- Product of zeroes = - d / a
Therefore
⇒ ( x ) ( y ) ( z ) = - d/a
⇒ ( a - d ) ( a ) ( a + d ) = - ( - 28 ) / 1
⇒ ( 4 - d ) ( 4 ) ( 4 + d ) = 28
⇒ ( 4 - d ) ( 4 + d ) ( 4 ) = 28
⇒ ( 16 - d² ) ( 4 ) = 28
⇒ ( 16 - d² ) = 28/4
⇒ ( 16 - d² ) = 7
⇒ 16 - 7 = d²
⇒ 9 = d²
⇒ √9 = d
⇒ ± 3 = d
Therefore ±3 is the correct answer.