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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Haya, tutatumia milinganyo uliyopewa kutatua kila sehemu. Milinganyo uliyopewa ni: 1. x+y=8 2. x^2+y^2=34 a) Tafuta thamani ya x^2+2xy+y^2 Step 1: Tambua usemi. Usemi x^2+2xy+y^2 ni sawa na (x+y)^2. Step 2: Badilisha thamani ya x+y. Tunajua kutoka kwa mlinganyo wa kwanza kwamba x+y=8. x^2+2xy+y^2 = (x+y)^2 = 8^2 Step 3: Rahisisha. 8^2 = 64 Thamani ya x^2+2xy+y^2 ni 64. b) Tafuta thamani ya 2xy Step 1: Tumia matokeo ya sehemu (a). Tunajua kwamba (x+y)^2 = x^2+2xy+y^2. Pia tunajua (x+y)^2 = 64 na x^2+y^2=34. Hivyo, tunaweza kuandika: 64 = (x^2+y^2) + 2xy Step 2: Badilisha thamani ya x^2+y^2. 64 = 34 + 2xy Step 3: Tatua kwa 2xy. Toa 34 kutoka pande zote mbili: 2xy = 64 - 34 2xy = 30 Thamani ya 2xy ni 30. c) Tafuta thamani ya x^2-2xy+y^2 Step 1: Tambua usemi. Usemi x^2-2xy+y^2 ni sawa na (x-y)^2. Pia tunaweza kuandika kama (x^2+y^2) - 2xy. Step 2: Badilisha thamani ya x^2+y^2 na 2xy. Tunajua x^2+y^2=34 na kutoka sehemu (b) tumepata 2xy=30. x^2-2xy+y^2 = 34 - 30 Step 3: Rahisisha. 34 - 30 = 4 Thamani ya x^2-2xy+y^2 ni 4. d) Tafuta thamani ya x-y Step 1: Tumia matokeo ya sehemu (c). Tunajua kwamba (x-y)^2 = x^2-2xy+y^2. Kutoka sehemu (c), tumepata x^2-2xy+y^2 = 4. Hivyo, (x-y)^2 = 4 Step 2: Chukua mzizi wa mraba pande zote mbili. x-y = ±sqrt(4) Step 3: Rahisisha. x-y = ± 2 Thamani ya x-y ni 2 au -2. Send me the next one 📸