1.Algebra Booster

2.2 Algebra Booster
fi
fi
fi
fi
2 2
2
x + 2 Ê Á b ˆ x + Ê b ˆ - Ê b ˆ + c = 0
Ë
˜
2a¯ Á
Ë
˜ Á ˜
2a¯ Ë2a¯
a
2 2
Ê b ˆ b - 4ac
Áx
+ =
Ë
˜
2
2a¯
4a
Ê b ˆ b - 4ac
Áx
+ = ±
Ë
˜
2a¯
2a
- b± b -4ac
x =
2a
2
2
Notes
1. The expression b 2 – 4ac is called the discriminant
and is denoted as D defined as D = b 2 – 4ac. The
value of D determines whether the quadratic equation
has two real solutions, one real solution and no
real solution according as D is positive (+ve), zero
and negative (–ve).
1.8 NATURE OF THE ROOTS
Let ax 2 + bx + c = 0, a π 0, whereas D = b 2 – 4ac.
The nature of the roots depends on D.
(i) If D > 0, the roots are real and distinct.
(ii) If D = 0, the roots are real and equal.
(iii) If D < 0, the roots are imaginary and distinct.
(iv) If D > 0 and a perfect square, the roots are rational.
(v) If D > 0 and not a perfect square, the roots are irrational.
(vi) If a, b, c ΠR and one of the roots is imaginary, say a +
ib, its other root will be its conjugate, i.e. a – ib.
(vii) If a, b, c ΠQ and one of the roots is irrational, say
p + q, its other root will be its conjugate, i.e. p - q.
(viii) If a = 1, b, c ΠQ and D = b 2 Р4ac is a perfect square,
both the roots are integers.
(ix) If a + b + c = 0 ( i.e. the sum of the co-efficients is
c
zero), 1 is one root and the other root will be .
a
c
(x) If a – b + c = 0, –1 is one root and the other root is - .
a
(xi) If the equation ax 2 + bx + c = 0 has real roots a and b,
we write
2
ax + bx + c = a( x -a)( x - b)
.
1.9 SUM AND PRODUCT OF THE ROOTS
(i) If a and b be the roots of ax 2 + bx + c = 0,
then
b
(a) a + b = Sum of the roots = -
a
c
(b) ab = product of the roots = .
a
(ii) If a, b, g are the roots of ax 3 + bx 2 + cx + d = 0,
then
b
(a) a + b + g = -
a
c
(b) ab + bg + ga =
a
d
(c) abg =-
a
(iii) If a, b, g, d are the roots of ax 4 + bx 3 + cx 2 + dx + c = 0,
then
b
(a) a + b + g + d = -
(b)
(c)
(d)
a
ab + ag + ad + bg + bd + gd =
d
abg + abd + agd + bgd = -
e
a
abgd =-
a
1.10 SYMMETIC FUNCTIONS OF THE ROOTS
In order to find the value of the symmetric function of the
roots a and b, we should express the given function in terms
of a + b and ab.
2
(i) ( a - b) = ( a + b) -4ab
(ii)
(iii)
(iv)
(v)
(vi)
2 2 2
a + b = ( a + b) -2ab
3 3 3
a + b = ( a + b) - 3 ab( a + b)
4 4 2 2 2 2
a + b = ( a + b ) -2( ab)
2 2 2
={( a + b) –2 ab} –2( ab)
5 5 3 3 2 2 2 2
a + b = ( a + b )( a + b )- a b ( a + b)
7 7 4 4 3 3 3 3
a + b = ( a + b )( a + b )- a b ( a + b)
1.11 FORMATION OF AN EQUATION
(i) If a and b be the roots of ax 2 + bx + c = 0,
then
2 Ê bˆ
c
x -Á- x+ = 0
Ë
˜
a¯
a
2
fi x -( a + b) x+ ab = 0.
(ii) If a, b and g are the roots of ax 3 + bx 2 + cx + d = 0, then
3 Ê bˆ
2 c d
x -Á- x + x - = 0
Ë
˜
a¯
a a
fi x 3 – (a + b + g)x 2 + (ab + bg + ga)x – abg = 0.
(iii) If a, b, g, d are the roots of ax 4 + bx 3 + cx 2 + dx + e = 0,
then
4 Ê bˆ 3 Êcˆ 2 Ê dˆ
e
x -Á- ˜ x + Á ˜ x -Á- ˜ x+ = 0 .
Ë a¯ Ëa¯ Ë a¯
a
4 3
fi x -( a + b + g + d)
x
2
+ ( ab + ag + ad + bg + bd + gd )x
c
a
-( abg + abd + agd + bgd ) x + abgd = 0
1.12 COMMON ROOTS OF QUADRATIC EQUATIONS
Let a 1
x 2 + b 1
x + c 1
= 0 and a 2
x 2 + b 2
x + c 2
= 0 be two quadratic
equations.
(i) When one root is common
Let a be a common root between the two equations. Then
a 1
a 2 + b 1
a + c 1
= 0
and a 2
a 2 + b 2
a + c 2
= 0