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
(a) (z-1)^2 - 3(z-3) Step 1: Expand (z-1)^2. (z-1)^2 = z^2 - 2z + 1 Step 2: Expand 3(z-3). 3(z-3) = 3z - 9 Step 3: Subtract. z^2 - 2z + 1 - (3z - 9) = z^2 - 2z + 1 - 3z + 9 = z^2 - 5z + 10 Simplified: z^2 - 5z + 10 z^2 - 5z + 10 (b) x^2 - 5x + 6 = 0 Step 1: Sum of roots formula: -(b)/(a). a=1, b=-5 -(-5)/(1) = 5 Step 2: Product of roots formula: (c)/(a). c=6 (6)/(1) = 6 Sum = 5, Product = 6 Sum = 5, Product = 6 (c) 2x^2 + 3x - 2 = 0 Step 1: Quadratic formula: x = -b ± sqrt(b^2 - 4ac)2a. a=2, b=3, c=-2 Step 2: Discriminant D = b^2 - 4ac. D = 3^2 - 4(2)(-2) = 9 + 16 = 25 sqrt(D) = 5 Step 3: Roots. x = (-3 ± 5)/(4) x_1 = (-3 + 5)/(4) = (2)/(4) = (1)/(2) x_2 = (-3 - 5)/(4) = (-8)/(4) = -2 Roots: x = (1)/(2), -2 x = (1)/(2), -2 (d) x^2 + 4x + 4 = 0 Step 1: Quadratic formula: x = -b ± sqrt(b^2 - 4ac)2a. a=1, b=4, c=4 Step 2: Discriminant D = b^2 - 4ac. D = 16 - 16 = 0 sqrt(D) = 0 Step 3: Root. x = (-4 ± 0)/(2) = (-4)/(2) = -2 Root: x = -2 (repeated) x = -2 (e) 3x^2 + 2x - 1 = 0 Step 1: Quadratic formula: x = -b ± sqrt(b^2 - 4ac)2a. a=3, b=2, c=-1 Step 2: Discriminant D = b^2 - 4ac. D = 4 - 4(3)(-1) = 4 + 12 = 16 sqrt(D) = 4 Step 3: Roots. x = (-2 ± 4)/(6) x_1 = (-2 + 4)/(6) = (2)/(6) = (1)/(3) x_2 = (-2 - 4)/(6) = (-6)/(6) = -1 Roots: x = (1)/(3), -1 x = (1)/(3), -1