Tiêu đề CH 1. Quadratic Equation (Math +1).pdf ✅

1. QUADRATIC EXPRESSION The general form of a quadratic expression in x is, f (x) = ax2 + bx + c, where a, b, c  R & a  0. and general form of a quadratic equation in x is, ax2 + bx + c = 0, where a, b, c  R & a  0. 2. ROOTS OF QUADRATIC EQUATION (a) The solution of the quadratic equation, ax2 + bx + c = 0 is given by x = 2 b ± b 4ac 2a   The expression D = b2 – 4ac is called the discriminant of the quadratic equation. (b) If   & are the roots of the quadratic equation ax2 + bx + c = 0, then ; (i)  +  = – b/a (ii) α β = c/a (iii) D | α β |= | a |  . (c) A quadratic equation whose roots are  &  is (x – ) (x – ) = 0 i.e. x2 – (+ ) x +  = 0 i.e. x2 – (sum of roots) x + product of roots = 0. 2 y (ax bx c) a(x ) (x )         2 b D a x 2a 4a          3. NATURE OF ROOTS (a) Consider the quadratic equation ax2 + bx + c = 0 where a, b, c  R & a  0 then; (i) D > 0  roots are real & distinct (unequal). (ii) D = 0  roots are real & coincident (equal). (iii) D < 0  roots are imaginary. (iv) If p + i q is one root of a quadratic equation, then the other must be the conjugate p – i q & vice versa. p, q R & i = -1   . (b) Consider the quadratic equation ax2 + bx + c = 0 where a, b, c  Q & a  0 then; (i) If D > 0 & is a perfect square, then roots are rational & unequal. (ii) If α = p + q is one root in this case, (where p is rational & q is a surd) then the other root must be the conjugate of it i.e. β = p - q & vice versa. Remember that a quadratic equation cannot have three different roots & if it has, it becomes an identity. QUADRATIC EQUATION QUADRATIC EQUATION 2