General Quadratic Equation: An equation of the form
ax2 + bx + c = 0
where a ¹ 0, is called a quadratic equation, in the real or complex coefficients a, b and c.
Roots of a Quadratic Equation:
The values of x (say x = a, ß) which satisfy the quadratic equation (1) are called the roots of the equation and they are given by
a = -b+ (a2 - 4ac) / 2a
ß = -b- (a2 - 4ac) / 2a
Discriminant of a Quadratic Equation: The quantity D = b2 - 4ac, is known as the discriminant of the equation.
Nature of the Roots: In quadratic equation ax2 + bx + c = 0, let us suppose that a, b, c are real and a ¹ 0. The following is true about the nature of its roots then
(i) The equation has real and distinct roots if and only if D = b2 - 4ac > 0.
(ii) The equation has real and coincident (equal roots if and only if D = b2 - 4ac = 0.
(iii) The equation has rational roots if and only if a, b, CEQ (the set of rational numbers) and D = b2 - 4ac is a perfect square (of a rational number).
(iv) The equation has (unequal) irrational (surd form) roots if and only if D = b2 - 4ac > 0 and not a perfect square even if a, band c are rational. In this case if p + q1/2, p, q rational is an irrational root, then p-q1/2 is also a root (a, b, c being rational).
(v) a + ib (b ¹ 0 and a, b Î R) is a root if and only if its conjugate a - ib is a root, that is complex roots always occur in conjugate pairs in a quadratic equation. In case the equation is satisfied by more than two complex numbers, then it reduces to an identity.
0. x2 + 0. x + 0 = 0, i.e., a = 0 = b = c.
Relation between Roots and Coefficients: If a, ß are the roots of the quadratic equation ax2 + bx + c = 0, then the sum and product of the roots is
a+ ß = -b/a and a ß = C/A
Hence the quadratic equation whose roots are a and ß is given by
x2 - (a + ß) x + aß = 0 or (x - a) (x - ß) = 0.
Condition that the Two Quadratic Equations have a Common Root:
Thus eliminating a., the condition for a common root is given by
(c1a2 - c2a1)2 = (b1c2 - b2c1) (a1b2- a2b1)
It is to be noted here 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: The two quadratic equations will have the same roots if and only if their coefficients are proportional, i.e.,
a1/a2 = b1/b2 = c1/c2
Condition that one root of a quadratic equation may be the square of the other root, i.e. the roots are a and ß = a2 is b3 + ca2 + ac2 = 3abc.
Higher Degree Equation: The equation
p(x) = a0xn + a1xn-1 + .......... + zn-1 x + an = 0
Where the coefficients a0, a1, ........, an Î R (or C) and a0 ¹ 0 is called an equation of nth degree, which has exactly n roots a1, a2, ... , an Î C.
Σa1 = a1, + a2 + ........ + an = -a1 / a0
Σa1 a2 = a1a2 + ........... + an-1, an = a0/a1
and so on and a1,a2 ......... an = (-1)n an/a0
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