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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Question 1 (a) Step 1: Apply the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a where a = 3, b = -8, c = 4. Step 2: Compute the discriminant = b^2 - 4ac = (-8)^2 - 4 · 3 · 4 = 64 - 48 = 16. Step 3: Take the square root sqrt() = sqrt(16) = 4. Step 4: Substitute the values x = (8 ± 4)/(2 · 3) = (8 ± 4)/(6). Step 5: Simplify x = (8 + 4)/(6) = (12)/(6) = 2, x = (8 - 4)/(6) = (4)/(6) = (2)/(3). x = 2,\ x = (2)/(3) Question 1 (b) Step 1: Apply the quadratic formula y = -b ± sqrt(b^2 - 4ac)2a where a = 4, b = -8, c = -3. Step 2: Compute the discriminant = b^2 - 4ac = (-8)^2 - 4 · 4 · (-3) = 64 + 48 = 112. Step 3: Simplify the discriminant = 16 · 7, sqrt() = 4sqrt(7). Step 4: Substitute the values y = 8 ± 4sqrt(7)2 · 4 = 8 ± 4sqrt(7)8 = 2 ± sqrt(7)2. y = 2 + sqrt(7)2,\ y = 2 - sqrt(7)2 Question 2 (a) Step 1: From x - y = 1, solve for y y = x - 1. Step 2: Substitute into the first equation sqrt(x^2 + (x - 1)^2) = 3. Step 3: Square both sides x^2 + (x - 1)^2 = 9. Step 4: Expand x^2 + (x^2 - 2x + 1) = 9 2x^2 - 2x + 1 = 9 2x^2 - 2x - 8 = 0. Step 5: Divide by 2 x^2 - x - 4 = 0. Step 6: Apply the quadratic formula x = 1 ± sqrt((-1)^2 - 4(1)(-4))2 · 1 = 1 ± sqrt(1 + 16)2 = 1 ± sqrt(17)2. Step 7: Find corresponding y values y = x - 1 = 1 ± sqrt(17)2 - 1 = 1 ± sqrt(17) - 22 = -1 ± sqrt(17)2. ( 1 + sqrt(17)2,\ -1 + sqrt(17)2 ),\ ( 1 - sqrt(17)2,\ -1 - sqrt(17)2 ) Question 2 (c) Step 1: Set y = 0 to find roots x^2 - 3x + 2 = 0. Step 2: Factor the quadratic (x - 1)(x - 2) = 0. Step 3: Solve x = 1,\ x = 2. The graph is a parabola opening upwards with vertex at ( (3)/(2), -(1)/(4) ) and two distinct real roots. x=1,\ x=2