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
Huu hapa ni ufumbuzi wa kutafuta thamani ya k: Step 1: Rahisisha maneno katika mlinganyo. Tunatumia sheria ya logariti _b (x^p) = p _b x kurahisisha _3 k^0.5: _3 k^0.5 = 0.5 _3 k Kisha, tunatathmini _2 8: Kwa kuwa 2^3 = 8, basi _2 8 = 3. Step 2: Badilisha maneno yaliyorahisishwa kwenye mlinganyo asilia. Mlinganyo unakuwa: (_3 k)^2 - 0.5 _3 k = 3 Step 3: Weka x = _3 k ili kuunda mlinganyo wa quadratic. x^2 - 0.5x = 3 Panga upya mlinganyo katika fomu ya kawaida ya quadratic ax^2 + bx + c = 0: x^2 - 0.5x - 3 = 0 Ili kuepuka desimali, zidisha mlinganyo mzima kwa 2: 2x^2 - x - 6 = 0 Step 4: Tatua mlinganyo wa quadratic kwa x. Tutatumia njia ya kuzidisha (factoring): 2x^2 - 4x + 3x - 6 = 0 2x(x - 2) + 3(x - 2) = 0 (2x + 3)(x - 2) = 0 Hii inatoa thamani mbili zinazowezekana kwa x: 2x + 3 = 0 2x = -3 x = -(3)/(2) x - 2 = 0 x = 2 Step 5: Badilisha x = _3 k na utatue kwa k. Kesi 1: x = 2 _3 k = 2 Kwa ufafanuzi wa logariti, k = 3^2. k = 9 Kesi 2: x = -(3)/(2) _3 k = -(3)/(2) k = 3^-(3)/(2) k = (1)/(3^3/2) k = (1)/(sqrt(3^3)) k = (1)/(sqrt(27)) k = (1)/(3sqrt(3)) Ili kurahisisha denominator, zidisha kwa sqrt(3)sqrt(3): k = (1)/(3sqrt(3)) × sqrt(3)sqrt(3) = sqrt(3)3 × 3 = sqrt(3)9 Thamani zote mbili za k ni chanya, hivyo zote ni suluhisho halali. Majibu ya k ni: k = 9 au k = sqrt(3)9 Tuma swali linalofuata 📸