This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
1) Find the nature of roots for the following quadratic equations a) 2x^2 - 5x + 2 = 0 Step 1: Identify a, b, c. 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 roots. Since = 9 > 0, the roots are real and distinct. The roots are real and distinct. b) 3x^2 - 7x + 4 = 0 Step 1: Identify a, b, c. 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 roots. Since = 1 > 0, the roots are real and distinct. The roots are real and distinct. 2) If and are the roots of the equation 3x^2 - 5x + 4 = 0, find the value of / + /. Step 1: Identify a, b, c and find the sum and product of roots. For 3x^2 - 5x + 4 = 0, a = 3, b = -5, c = 4. Sum of roots: + = -(b)/(a) = -(-5)/(3) = (5)/(3). Product of roots: = (c)/(a) = (4)/(3). Step 2: Simplify the expression ()/() + ()/(). ()/() + ()/() = (^2 + ^2)/() Step 3: Use the identity ^2 + ^2 = ( + )^2 - 2. ^2 + ^2 = ((5)/(3))^2 - 2((4)/(3)) = (25)/(9) - (8)/(3) = (25)/(9) - (24)/(9) = (1)/(9) Step 4: Substitute the values into the simplified expression. (^2 + ^2)/() = (1)/(9)(4)/(3) = (1)/(9) × (3)/(4) = (3)/(36) = (1)/(12) The value is (1)/(12). 3) If and are the root of the equation 2x^2 - 4x + 7 = 0, find ^2 + ^2. Step 1: Identify a, b, c and find the sum and product of roots. For 2x^2 - 4x + 7 = 0, a = 2, b = -4, c = 7. Sum of roots: + = -(b)/(a) = -(-4)/(2) = 2. Product of roots: = (c)/(a) = (7)/(2). Step 2: Use the identity ^2 + ^2 = ( + )^2 - 2. ^2 + ^2 = (2)^2 - 2((7)/(2)) = 4 - 7 = -3 The value is -3. 4) Construct and simplify equations whose roots are -3, 1. Step 1: Find the sum of the roots. Sum of roots = -3 + 1 = -2. Step 2: Find the product of the roots. Product of roots = (-3)(1) = -3. Step 3: Construct the quadratic equation using the form x^2 - (sum of roots)x + (product of roots) = 0. x^2 - (-2)x + (-3) = 0 x^2 + 2x - 3 = 0 The equation is x^2 + 2x - 3 = 0.