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
Step 1: Tumia sheria ya logariti a - b = ((a)/(b)) kurahisisha pande zote mbili za mlinganyo. ((2x - 11)/(2)) = ((3)/(x)) Step 2: Kwa kuwa A = B inamaanisha A = B, linganisha hoja za logariti. (2x - 11)/(2) = (3)/(x) Step 3: Tatua mlinganyo wa aljebra kwa x. Zidisha pande zote mbili kwa 2x kuondoa vigawe. x(2x - 11) = 2 × 3 2x^2 - 11x = 6 Panga upya mlinganyo kuwa fomu ya quadratic ax^2 + bx + c = 0. 2x^2 - 11x - 6 = 0 Step 4: Tatua mlinganyo wa quadratic kwa kutumia njia ya kuzidisha. Tafuta namba mbili zinazozidisha hadi 2 × (-6) = -12 na kujumlisha hadi -11. Namba hizo ni -12 na 1. 2x^2 - 12x + x - 6 = 0 Toa 2x kama kigezo cha kawaida kutoka kwa maneno mawili ya kwanza na 1 kutoka kwa maneno mawili ya mwisho. 2x(x - 6) + 1(x - 6) = 0 Toa (x - 6) kama kigezo cha kawaida. (2x + 1)(x - 6) = 0 Hii inatoa suluhisho mbili zinazowezekana: 2x + 1 = 0 2x = -1 x = -(1)/(2) x - 6 = 0 x = 6 Step 5: Angalia uhalali wa suluhisho. Hoja ya logariti lazima iwe chanya. Kwa x = -(1)/(2): 2x - 11 = 2(-(1)/(2)) - 11 = -1 - 11 = -12. Hii ni hasi, hivyo (-12) haijafafanuliwa. x = -(1)/(2). Hii ni hasi, hivyo (-(1)/(2)) haijafafanuliwa. Kwa hiyo, x = -(1)/(2) si suluhisho halali. Kwa x = 6: 2x - 11 = 2(6) - 11 = 12 - 11 = 1. Hii ni chanya. x = 6. Hii ni chanya. Hoja zote za logariti ni chanya, hivyo x = 6 ni suluhisho halali. Thamani ya x inayokidhi mlinganyo ni: x = 6 Tuma swali linalofuata 📸