QUADRATIC EQUATIONS
Quadratic Equation : A general quadratic equation is expressed as ax2 + bx + c = 0
where a ¹ 0, a, b and c may be real or complex coefficients.
Roots of the quadratic equation
A quadratic equation has two roots a and b given by
a = -b + Öb2 – 4ac
2a
and b =
-b – Öb2 – 4ac 2a
The quantity D = b2 – 4ac is known as the discriminant of the equation.
- If D = b2 – 4ac > 0 the roots are real and distinct
- If D = b2 – 4ac = 0 the roots are real and equal
- If D = b2 – 4ac < 0 the roots are
Relation between roots and coefficients : If a and b are the roots of the equa- tion ax2 + bx + c = 0 then
a + b = – b and ab = c
a a
Hence
x2 – (a + b) x + ab = 0 or (x – a) (x – b) = 0
Condition that the Two Quadratic Equations have a Common Root : Sup-
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pose we have two quadratic equation a x2 + b x + c = 0 and a x2 + b x + c
= 0, then
eliminating a, the condition for a common root is given by
(c a
- c a )2 = (b c
- b c ) (a b
- a b )
1 2 2 1
1 2 2 1
1 2 2 1
Here one thing is to be noted, that two different quadratic equations with rational coefficients cannot have a common root which is non-real complex or irrational, as imaginary and surd roots always occur in pairs.
Condition that the Two Quadratic Equations have both the Roots Common
: If we consider the two above quadratic equations, they will have the same roots if and only if their coefficients are proportional, i.e.,
a1 = b1 = c1
a2 b2 c2
Condition that one root of a quadratic equation may be the square of the other root, i.e. the roots are a and b = a2 is b3 + ca2 + ac2 = 3abc.
Higher Degree Equation : The equation of higher degree
p(x) º a xn + a xn-1 + … + a x + a = 0
0 1 n-1 n
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Where the coefficinets a ,a ,…,a Î R (or C) and a
¹ 0 is called an equation of
nth degree, which has exactly n roots a , a , …, a Î C.
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Sa = a + a
+…+ a
1 2 n
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= a1 a0
Sa a
= a a +…+a
a = a1 and so on and a a …, a
= (-1)n
an .
1 2 1 2
n-l n a0
1 2 n a0
