QUADRATIC EQUATION - Concept, Properties with Example
The algebraic expression of the form ax2 + bx + c, a 0 is called a quadratic expression, because the highest order term in it is of second degree. Quadratic equation means, ax2 + bx + c = 0. In general whenever one says zeroes of the expression ax2+bx + c, it implies roots of the equation ax2 + bx + c = 0, unless specified otherwise.
A quadratic equation has exactly two roots which may be real (equal or unequal) or imaginary.
SOLUTION OF QUADRATIC EQUATION & RELATION BETWEEN ROOTS & CO-EFFICIENTS
- The general form of quadratic equation is
The roots can be found in following manner
This expression can be directly used to find the two roots of a quadratic equation.
- The expression b2 – 4 ac = D is called the discriminant of the quadratic equation
- if are the roots of the quadratic equation then
Nature of Roots :
(a). Consider the quadratic equation ax2 + bx + c = 0 where a,b,c are real number a!=0 then
- D > 0 => roots are real & distinct (unequal).
- D = 0 => roots are real & coincident (equal).
- D < 0 => roots are imaginary.
- if p + i q is one root of a quadratic equation, then the other root must be the conjugate p–iq&vice versa.
Find all the integral values of a for which the quadratic equation (x – a)(x – 10) + 1 = 0 has integral roots.
Here the equation is
Since integral roots will always be rational it means D should be a perfect square.
If D is a perfect square it means we want difference of two perfect square as 4 which is possible only when
(a–10)(a-10) = 4 and D = 0.
(a-10) = +2 or (a-10) = -2
value of a should be 12, 8
ROOTS UNDER PARTICULAR CASES