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 the problems: 1. Find the sum and product of the roots of the following, then (briefly) state the roots: a) x^2 - 5x + 6 = 0 For a quadratic equation ax^2 + bx + c = 0: Here, a=1, b=-5, c=6. Step 1: Calculate the sum of the roots. Sum of roots = -(b)/(a) = -(-5)/(1) = 5 Step 2: Calculate the product of the roots. Product of roots = (c)/(a) = (6)/(1) = 6 Step 3: Determine the nature of the roots using the discriminant D = b^2 - 4ac. D = (-5)^2 - 4(1)(6) = 25 - 24 = 1 Since D > 0, the roots are real and distinct. Sum of roots: 5* Product of roots: 6* Nature of roots: Real and distinct* b) 3x^2 - 12x + 12 = 0 For a quadratic equation ax^2 + bx + c = 0: Here, a=3, b=-12, c=12. Step 1: Calculate the sum of the roots. Sum of roots = -(b)/(a) = -(-12)/(3) = 4 Step 2: Calculate the product of the roots. Product of roots = (c)/(a) = (12)/(3) = 4 Step 3: Determine the nature of the roots using the discriminant D = b^2 - 4ac. D = (-12)^2 - 4(3)(12) = 144 - 144 = 0 Since D = 0, the roots are real and equal. Sum of roots: 4* Product of roots: 4* Nature of roots: Real and equal* 2. The sum and product of the roots of a quadratic equation are 4 and -5 respectively. Form the quadratic equation. Let the sum of the roots be S and the product of the roots be P. Given S = 4 and P = -5. The general form of a quadratic equation is x^2 - Sx + P = 0. Step 1: Substitute the given sum and product into the general form. x^2 - (4)x + (-5) = 0 Step 2: Simplify the equation. x^2 - 4x - 5 = 0 The quadratic equation is x^2 - 4x - 5 = 0. 3. If one root of a quadratic equation is -1 and the sum of the roots is 6, find: Let the two roots be and . Given one root, = -1. Given the sum of the roots, + = 6. a. The other root Step 1: Substitute the known root into the sum of roots equation. -1 + = 6 Step 2: Solve for . = 6 + 1 = 7 The other root is 7. b. The product of the roots Step 1: Multiply the two roots. Product = = (-1)(7) Product = -7 The product of the roots is -7. c. The quadratic equation We have the sum of roots S = 6 and the product of roots P = -7. The general form of a quadratic equation is x^2 - Sx + P = 0. Step 1: Substitute the sum and product into the general form. x^2 - (6)x + (-7) = 0 Step 2: Simplify the equation. x^2 - 6x - 7 = 0 The quadratic equation is x^2 - 6x - 7 = 0.