Find the nature of roots for quadratic equations and calculate expressions involving roots.

1) a) Find the nature of roots for 2x^2 - 5x + 2 = 0. Step 1: Identify the coefficients. For the quadratic equation ax^2 + bx + c = 0, we have a = 2, b = -5, c = 2. Step 2: Calculate the discriminant = b^2 - 4ac. = (-5)^2 - 4(2)(2) = 25 - 16 = 9 Step 3: Determine the nature of the roots. Since = 9 > 0, the roots are real and distinct. The roots are also rational because the discriminant is a perfect square. The nature of the roots is real and distinct. b) Find the nature of roots for 3x^2 - 7x + 4 = 0. Step 1: Identify the coefficients. For the quadratic equation ax^2 + bx + c = 0, we have a = 3, b = -7, c = 4. Step 2: Calculate the discriminant = b^2 - 4ac. = (-7)^2 - 4(3)(4) = 49 - 48 = 1 Step 3: Determine the nature of the roots. Since = 1 > 0, the roots are real and distinct. The roots are also rational because the discriminant is a perfect square. The nature of the roots is real and distinct. 2) If and are the roots of the equation 3x^2 - 5x + 4 = 0, find the value of ()/() + ()/(). Step 1: Identify the coefficients. For the quadratic equation ax^2 + bx + c = 0, we have a = 3, b = -5, c = 4. Step 2: Find the sum and product of the roots. Sum of roots: + = -(b)/(a) + = -(-5)/(3) = (5)/(3) Product of roots: = (c)/(a) = (4)/(3) Step 3: Simplify the expression ()/() + ()/(). ()/() + ()/() = (^2 + ^2)/( ) We know that ^2 + ^2 = ( + )^2 - 2 . So, the expression becomes: (( + )^2 - 2 )/( ) Step 4: Substitute the values of + and into the simplified expression. ((5)/(3))^2 - 2((4)/(3))(4)/(3) (25)/(9) - (8)/(3)(4)/(3) To subtract the terms in the numerator, find a common denominator: (25)/(9) - (8 × 3)/(3 × 3)(4)/(3) (25)/(9) - (24)/(9)(4)/(3) (1)/(9)(4)/(3) To divide by a fraction, multiply by its reciprocal: (1)/(9) × (3)/(4) (3)/(36) (1)/(12) The value of ()/() + ()/() is (1)/(12).