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.
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Here are the solutions to problems 1, 2, 3, and 4. 1) Find the nature of roots for the following quadratic equations To determine the nature of the roots of a quadratic equation ax^2 + bx + c = 0, we use the discriminant = b^2 - 4ac. If > 0, the roots are real and distinct. If = 0, the roots are real and equal. If < 0, the roots are complex (not real). a) 2x^2 - 5x + 2 = 0 Step 1: Identify the coefficients. Here, 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. b) 3x^2 - 7x + 4 = 0 Step 1: Identify the coefficients. Here, 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. 2) If and are the roots of the equation 3x^2 - 5x + 4 = 0, find the value of / + /. Step 1: Identify the coefficients of the quadratic equation ax^2 + bx + c = 0. For 3x^2 - 5x + 4 = 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. Substitute this into the expression: (^2 + ^2)/() = (( + )^2 - 2)/() Step 4: Substitute the values of ( + ) and . = ((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) Step 5: Perform the division. = (1)/(9) × (3)/(4) = (3)/(36) = (1)/(12) The value of ()/() + ()/() is (1)/(12). 3) If and are the roots of the equation 2x^2 - 4x + 7 = 0, find ^2 + ^2. Step 1: Identify the coefficients of the quadratic equation ax^2 + bx + c = 0. For 2x^2 - 4x + 7 = 0, we have a=2, b=-4, c=7. Step 2: Find the sum and product of the roots. Sum of roots: + = -(b)/(a) = -(-4)/(2) = (4)/(2) = 2. Product of roots: = (c)/(a) = (7)/(2). Step 3: Use the identity for ^2 + ^2. ^2 + ^2 = ( + )^2 - 2 Step 4: Substitute the values of ( + ) and . ^2 + ^2 = (2)^2 - 2((7)/(2)) ^2 + ^2 = 4 - 7 ^2 + ^2 = -3 The value of ^2 + ^2 is -3. 4) Construct and simplify equations whose roots are -3, 1. Step 1: Let the given roots be r_1 = -3 and r_2 = 1. Step 2: Calculate the sum of the roots (S). S = r_1 + r_2 = -3 + 1 = -2 Step 3: Calculate the product of the roots (P). P = r_1 × r_2 = (-3) × (1) = -3 Step 4: Form the quadratic equation using the general form x^2 - Sx + P = 0. x^2 - (-2)x + (-3) = 0 x^2 + 2x - 3 = 0 The constructed quadratic equation is x^2 + 2x - 3 = 0.