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: Andika upya mlinganyo na ueleze base ya logariti. Mlinganyo uliotolewa ni: ( n)^2 = _2 n - _2 8 Katika muktadha wa maswali ya hisabati ya KCSE, n bila base maalum inamaanisha _10 n. Hivyo, mlinganyo unakuwa: (_10 n)^2 = _2 n - _2 8 Step 2: Rahisisha neno la _2 8. Tunajua kuwa 8 = 2^3. Hivyo, _2 8 = _2 (2^3) = 3 Step 3: Badilisha thamani iliyorahisishwa kwenye mlinganyo. Mlinganyo unakuwa: (_10 n)^2 = _2 n - 3 Step 4: Badilisha _2 n kuwa base 10. Tumia formula ya kubadilisha base: _b a = (_c a)/(_c b). Hivyo, _2 n = _10 n_10 2. Mlinganyo unakuwa: (_10 n)^2 = _10 n_10 2 - 3 Step 5: Weka x = _10 n ili kuunda mlinganyo wa quadratic. Badilisha x kwenye mlinganyo: x^2 = (x)/(_10) 2 - 3 Panga upya mlinganyo ili uwe katika mfumo wa kawaida wa quadratic ax^2 + bx + c = 0: x^2 - (1)/(_10) 2 x + 3 = 0 Step 6: Tatua mlinganyo wa quadratic kwa x. Kwanza, hesabu thamani ya _10 2: _10 2 ≈ 0.30103 Sasa, badilisha thamani hii kwenye mlinganyo: x^2 - (1)/(0.30103) x + 3 = 0 x^2 - 3.3219 x + 3 = 0 Tumia formula ya quadratic: x = -b ± sqrt(b^2 - 4ac)2a. Katika mlinganyo huu, a=1, b=-3.3219, na c=3. Hesabu kibaguzi (discriminant), = b^2 - 4ac: = (-3.3219)^2 - 4(1)(3) = 11.0350 - 12 = -0.9650 Kwa kuwa kibaguzi () ni hasi (chini ya sifuri), hakuna suluhisho halisi (real solutions) kwa x. Hii inamaanisha kuwa hakuna thamani halisi ya _10 n inayoweza kukidhi mlinganyo huu, na kwa hivyo, hakuna thamani halisi ya n. Hakuna suluhisho halisi kwa n. Hakuna suluhisho halisi kwa n 3 done, 2 left today. You're making progress.