32 Chapter 2
Exercise 2.4
Q4iii). If α, β be roots of 2x 2 + 3x + 1 =
0 ,
x 2- ( Sumoftheroots ) x + ( Productoftheroots ) = 0 then find value of α 2 β +
β α 2
x 2 - ⎛ │ ⎝ Sol: Comparing
2x 2 + 3x + 1 =0 ⇒ 15x - 215
26 - ( ⎞ │ ⎠
- x 26 + )
( x 1 )
+ = 15 0 =
0
with the quadratic equation
ax 2 + bx + c =
0 ⇒ 15x 2+ 26x + 15 =
0 we have a = 2 , b = 3, c =
1
Q6. If α, β be roots of x 2- 4x + 2 =0 , then Since α,β be the roots of
2x 2 + 3x + 1 = 0 α + β = - a b = - 2 3
and α . β = c a =
2 1 Now α 2 β + β α 2 = α α . α β 2 +
β 2 α .
β β
=
α 3 αβ
+ β
3
find equation whose roots are
α + α 1 , β + β 1 Sol: Comparing x 2- 4x + 2 =0 with the quadratic equation
ax 2 + bx + c = 0 we have a = 1 , b = - 4, c =
2 Since α,β be the roots of ax 2 + bx + c = 0 =
( α + β ) 3 - αβ 3
αβ ( α + β ) ∴ ( a + b ) 3 = a 3 + b 3 + 3ab ( a +
b ) Putting values
α + β = - a b = - ( - )4 1 = 4 1 =
4 and
α . β = c a = 2 1 = 2 The roots of the required equation are
α 2 β + β α
2
= ⌈ │ │ ⌊ ⎛ │ ⎝ - ⎞ │ ⎠ 3 - ⎛ │ ⎝ ⎞⎛ ││ ⎠⎝ 3 2 3 1 2 - ⎞ │ ⎠ ⌉ │ │ ⌋
÷ α + α 1 , β + β 1 α 2 β + β
α
2
= ⌈ │ ⌊ 3 1 2 2
- 27 8 + 9 4 × 2 2 ⌉ │ ⌋
× 2 1
∴ Sum of the roots
1 1 1 1
α 2 β + β α
2
= - 27 8 + 18 ×
2 1 α β
α + α + β + β = α + β + β β α +
β 1 1
Putting the values
β α
3x 2 - 2x + 5 = 0 3x 2 - 2x + 5 =0 ∴ Product of the roots
ax 2 + bx + c = 0 a = b = c =
ax 2 + bx + c = 0 α + β = - a b = - ( - )2 3 = 2
3 α . β = c a =
5 3 Putting the values
α α
α + α + β + β = α + β +
β αβ + α 2 + 2 =
- 4 9Q5. If α, β be roots of , then find the equation whose roots are αβ , α
β 1 1 4 4 2
1 1 4 2 6
Sol: Comparing with the quadratic equation we have 3 , - 2, 5 Since α,β be the roots of
and
The roots of the required equation are αβ , α β ∴ Sum of the roots
2 2 3 2 5 3 5 3
4 30 3 9 5 26 15
α + α + β + β
= +
α + α + β + β
= + =
⎛ │ ⎝ α + α 1 ⎞⎛ ││ ⎠⎝ β + β 1 ⎞ │ ⎠
= αβ + α α α β + α β .
β β + αβ 1 ⎛ │ ⎝ α + α 1 ⎞⎛ ││ ⎠⎝ β + β 1 ⎞ │ ⎠
= αβ + αβ 1
+ α 2 αβ + β 2⎛ │ ⎝ α + α 1 ⎞⎛ ││ ⎠⎝ β + β 1 ⎞ │ ⎠ = αβ + αβ 1+ ( α + β )
2 - 2 αβ αβ
α β α α β β β α α β α β
1 1 2 1 2 ( 4 ) 2 2 ( 2 )
2 α β
αβ
1 1 2 2 .2 1 1 16 4 2 2 α β αβ αβ
1 1 4 1 16 4 17 αβ
2 2
α β αβ
αβ ⎛ │ ⎝ + = . +.
+ ⎞⎛ ││ ⎠⎝ + ⎞ │ ⎠ = + + - =
+ ⎛ │ ⎝ + ⎞⎛ ││ ⎠⎝ + ⎞ │ ⎠ = + + - =
+ + 2 - 2
⎛ │ ⎝ + ⎞⎛ ││ ⎠⎝ + ⎞ │ ⎠ = + + - = =
+ - 2
x 2- ( Sumoftheroots ) x + ( Productoftheroots ) =
0 2
( ) 2
α β . α β
= 1 α α β β 2 2
α α β β 2 2α α β β
2
x - 6 x + ⎛ │ ⎝ α β + α
β
= ⌈ │ │ ⌊ 17 2 ⎛ │ ⎝ ⎞ │ ⎠
= 0 ⇒ 2x - 12x + 17 = 0 ⎞ │ ⎠ - ⎛ │ ⎝ ⎞ │ ⎠ ⌉ │ │ ⌋ ÷ α β + α
β
= ⌈ │ ⌊ - ⌉ │ ⌋ × α β + α
β
= - ∴ Product of the roots so
× its General Formula
Divident = ( Divisor )( Quotient ) + Re mainder This result is known as “Division Algorithm”
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