If the value of real number α > 0 for which x 2 - 5 α x + 1 = 0 and x 2 - α x - 5 = 0 have a common real roots is 3 2 β then β is equal to ________

Step-by-step solution

Given, Quadratic equation x 2 - 5 α x + 1 = 0 and x 2 - α x - 5 = 0 have a common real root, So, taking k to be common root then equation becomes, k 2 - 5 α k + 1 = 0   . . . . . . 1 k 2 - α k - 5 = 0   . . . . . 2 Now subtraction above equations we get, k = 6 4 α Now putting the value of k in equation 1 we get, 6 4 α 2 - 5 α 6 4 α + 1 = 0   ⇒ 6 4 α 2 - 26 4 = 0   ⇒ 36 16 α 2 = 26 4 ⇒ α 2 = 9 26 ⇒ α = 3 2 ×