Algebra DPP 1.3 Quadratics
Q1 . For what values of b do the equations: 1988x² + bx + 8891 = 0 and 8891x² + bx + 1988 = 0 have a common root? Solution: We are given two quadratic equations: 1988 x 2 + b x + 8891 = 0 1988x^2 + bx + 8891 = 0 8891 x 2 + b x + 1988 = 0 8891x^2 + bx + 1988 = 0 We are to find the values of b b for which they have a common root . Step 1: Let α \alpha be the common root. Then α \alpha satisfies both equations. So, From equation (1): 1988 α 2 + b α + 8891 = 0 (i) 1988\alpha^2 + b\alpha + 8891 = 0 \quad \text{(i)} From equation (2): 8891 α 2 + b α + 1988 = 0 (ii) 8891\alpha^2 + b\alpha + 1988 = 0 \quad \text{(ii)} Step 2: Subtract equation (i) from equation (ii): ( 8891 α 2 − 1988 α 2 ) + ( b α − b α ) + ( 1988 − 8891 ) = 0 (8891\alpha^2 - 1988\alpha^2) + (b\alpha - b\alpha) + (1988 - 8891) = 0 ( 8891 − 1988 ) α 2 − ( 8891 − 1988 ) = 0 (8891 - 1988)\alpha^2 - (8891 - 1988) = 0 6903 α 2 − 6903 = 0 6903\alpha^2 - 6903 = 0 α 2 = 1 ⇒ α = ± 1 \alpha^2 = 1 \Rightar...