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Title: Algebra _ Relation between roots and coefficients
Description: Algebra _ Relation between roots and coefficients Solution: Let α and β be the roots of the given equation. Sometimes the relation between roots of a quadratic equation is given and we are asked to find the condition i.e., relation between the coefficients a, b and c of quadratic equation. This is easily done using the formula α + β = -ba and αβ = ca.
Description: Algebra _ Relation between roots and coefficients Solution: Let α and β be the roots of the given equation. Sometimes the relation between roots of a quadratic equation is given and we are asked to find the condition i.e., relation between the coefficients a, b and c of quadratic equation. This is easily done using the formula α + β = -ba and αβ = ca.
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Algebra
Relation between roots and coefficients
1
...
Form the equation whose roots are
(a)
3, 4
Solution:
The roots are 3 and 4
Sum of the roots
= 7
Product of the roots
= 12
The required equation isx2 – 7x + 12
(b)
= 0
3 + –7 , 3 – –7
Solution:
The roots are 3 +– 7 and 3 – – 7
Sum of the roots
Product of the roots
= 6
= 9–7 = 2
The required equation is x2 – 6x + 2 = 0
(c)
4 + –2 ,
2
4 – –2
2
Solution:
The roots are 4 + –2
2
and
Sum of the roots
=
Product of the roots
=
4 – –2
2
4 + –2
4 – –2
+
2
2
4 +–2
2
The required equation is x2 – 4x + 7 = 0
2
2x2 – 8x + 7 = 0
4 ––2
2
8
2
=
=
= 4
7
16 – 2
=
2
4
3
...
Solution:
+
2 + 2
6
3
=
=
= 2
4
3
= ( + )2 2
4
3
= (2)2 – 2
4
...
+ =
Solution:
+
=
=
5
2
12
2
= 6
2 + 2
5
2
= 4 8 =
3
=
2
+ 2 (6)
=
6
=
73
24
( + )2 – 2
25
+ 12
4
=
6
5
...
Find the quadratic equation
...
= 16
= 8
The equation is x2 8x + 0
x2 8x
= 0
= 0
Hence = 0
= 0
Form the equation one of whose roots is – 5 and sum of the roots is –2
...
If and are the roots of the equation x2 – 3x – 4 = 0 form the equation
whose roots are
1
1
2 , 2
(b)
x2 – 3x – 4
= 0
(a)
Solution:
+
(a)
Sum =
,
...
x
4
x+1 = 0
4x2 + 17x + 4
= 0
If and are the roots of the equation x2 x = 15 form the equation whose
roots are ( + ) and 3
...
= 0
If and are the roots of the equation x2 2x + 7 = 0 form the equation whose
roots are 2 , 2
...
2
= 3 3
= ( )3 = (7)3
= 343
The equation is x2 – 14x + 343
= 0
10
...
Solution:
Let the roots be and 2
+ 2
=
a
a
a
3 = 2 = 6
2
Also
...
= 4 x 6 = 24
If one root of the equation 3x2 + kx – 81 = 0 is the square of the other, find k
...
2
k
3
s
81
=
3 = 27
3
= 27
+ 2
= 0
=
Substitute in (1)
3+9
=
k
3
6
=
k
3
= 3
k = 18
12
...
Solution:
Adding
p
5
+
=
=
2
=
p
+ 1
5
=
p+5
10
& =
= p + 5 1 =
10
Hence
p+5
10
=
p2 25
100
=
p2 – 25
p2
p
1
5
= 20
= 45
= 3 5
=
p–5
10
p–5
10
=
1
5
1
5
Title: Algebra _ Relation between roots and coefficients
Description: Algebra _ Relation between roots and coefficients Solution: Let α and β be the roots of the given equation. Sometimes the relation between roots of a quadratic equation is given and we are asked to find the condition i.e., relation between the coefficients a, b and c of quadratic equation. This is easily done using the formula α + β = -ba and αβ = ca.
Description: Algebra _ Relation between roots and coefficients Solution: Let α and β be the roots of the given equation. Sometimes the relation between roots of a quadratic equation is given and we are asked to find the condition i.e., relation between the coefficients a, b and c of quadratic equation. This is easily done using the formula α + β = -ba and αβ = ca.