| The trinomial expression of the form ax2 + bx + c = 0, a ≠ 0 a, b, c are R is called quadratic equation, because the highest order term in it is of second degree. We can say… (a) ax2 + b x + c = 0, a ≠ 0 has exactly two roots which may be real or unequal or equal or complex. (b) If ax2 + b x + c = 0, a ≠ 0 can not have three or more roots & if it has, it becomes an identity. If ax2 + b x + c = 0 is an identity then a = b = c = 0. SOLUTION OF Q.E. Let α and β be the roots of the equation ax2 + b x + c = 0, a ≠ 0. then (a) x1 & x2 = [-b +-(b2 - 4ac)1/2] / 2a(b) b2 - 4ac= D is called Discriminant of the Q.E. (d) x2 – (sum of roots) x + (product of roots) = 0 NATURE OR TYPE OF ROOTS D and α , β are the roots of the ax2 + b x + c = 0, a ≠ 0 If D greater than; 0 then roots are real, distinct.If D = 0 then roots are real, equal. If D less than 0 then roots are imaginary.If p + iq is one root of equation, then the other root is p – iq(conjugate of each other). If D is perfect square then the roots are rational.If p + √q is one root of equation, then the other root is p – √q. SIGN OF ROOTS OF Q.E : conditions for sign of roots (a) For Both roots are +ive then (i) D greater than 0(ii) sum of roots > 0 (iii) product of roots greater than 0 , these three conditions simultaneous true. (b) For Both roots are –ive (i) D greater than 0 (ii) sum of roots less than 0 (iii) product of roots greater than 0 , these three conditions simultaneous true. (c) For one root is +ive & one is –ive (i) D greater than 0 (ii) product of roots less than 0, these two conditions simultaneous true. |