Solve 9x2 + 16 = 24x, A passenger train has n passengers on a certain trip. Each of the passengers paid a fare of Ksh (2n+20) and the total fare collected was Ksh. 120,000. Determine the number of passengers. Locate the centre of enlargement and determine the linear scale factor.

Question

Asked on June 15, 2026Mathematics

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.

Accepted Answer · 2026-06-15T18:33:50.290Z

Haya hapa ni masuluhisho ya maswali yako: 1. Tatua 9x^2 + 16 = 24x. Step 1: Panga upya mlinganyo katika mfumo wa kawaida wa mlinganyo wa quadratic, ax^2 + bx + c = 0. 9x^2 - 24x + 16 = 0 Step 2: Tambua kwamba huu ni trinomial kamili ya mraba. (3x)^2 - 2(3x)(4) + (4)^2 = 0 (3x - 4)^2 = 0 Step 3: Tatua kwa x. 3x - 4 = 0 3x = 4 x = (4)/(3) Jibu la mwisho: x = (4)/(3) 2. Treni ya abiria ina abiria n katika safari fulani. Kila abiria alilipa nauli ya Ksh (2n+20) na jumla ya nauli iliyokusanywa ilikuwa Ksh 120,000. Bainisha idadi ya abiria. Step 1: Unda mlinganyo unaowakilisha jumla ya nauli iliyokusanywa. Jumla ya nauli = (Idadi ya abiria) × (Nauli kwa kila abiria) 120,000 = n(2n + 20) Step 2: Panua na panga upya mlinganyo katika mfumo wa kawaida wa quadratic. 120,000 = 2n^2 + 20n 2n^2 + 20n - 120,000 = 0 Step 3: Rahisisha mlinganyo kwa kugawanya pande zote mbili kwa 2. n^2 + 10n - 60,000 = 0 Step 4: Tumia fomula ya quadratic, n = -b ± sqrt(b^2 - 4ac)2a, kutatua kwa n. Hapa, a=1, b=10, c=-60,000. n = -10 ± sqrt(10^2 - 4(1)(-60,000))2(1) n = -10 ± sqrt(100 + 240,000)2 n = -10 ± sqrt(240,100)2 n = (-10 ± 490)/(2) Step 5: Kokotoa thamani zinazowezekana za n. n_1 = (-10 + 490)/(2) = (480)/(2) = 240 n_2 = (-10 - 490)/(2) = (-500)/(2) = -250 Kwa kuwa idadi ya abiria haiwezi kuwa hasi, tunachukua thamani chanya. Jibu la mwisho: Idadi ya abiria ni 240. 3. Tafuta kituo cha ukuzaji na bainisha kipimo cha ukuzaji (linear scale factor). Step 1: Tafuta kituo cha ukuzaji. Kituo cha ukuzaji hupatikana kwa kuchora mistari iliyonyooka inayounganisha ncha zinazolingana za umbo asili (QRS) na umbo lililokuzwa (Q'R'S'). Kwa mfano, unganisha Q na Q', R na R', na S na S'. Sehemu ambapo mistari hii inakutana ndio kituo cha ukuzaji. Kutoka kwenye mchoro, sehemu hii imeandikwa kama O. Kituo cha ukuzaji ni O. Step 2: Bainisha kipimo cha ukuzaji (linear scale factor). Kipimo cha ukuzaji (k) ni uwiano wa urefu wa upande wa umbo lililokuzwa kwa urefu wa upande unaolingana wa umbo asili. Kutokana na mahesabu yaliyotolewa, tunatumia uwiano wa 8.7 kwa 3. k = Urefu wa upande wa umbo lililokuzwaUrefu wa upande wa umbo asili k = (8.7)/(3) k = 2.9 Jibu la mwisho: Kituo cha ukuzaji ni O na kipimo cha ukuzaji ni 2.9. Tuma swali linalofuata 📸

Accepted Answer · 2026-06-15T18:33:50.290Z

Haya hapa ni masuluhisho ya maswali yako: 1. Tatua 9x^2 + 16 = 24x. Step 1: Panga upya mlinganyo katika mfumo wa kawaida wa mlinganyo wa quadratic, ax^2 + bx + c = 0. 9x^2 - 24x + 16 = 0 Step 2: Tambua kwamba huu ni trinomial kamili ya mraba. (3x)^2 - 2(3x)(4) + (4)^2 = 0 (3x - 4)^2 = 0 Step 3: Tatua kwa x. 3x - 4 = 0 3x = 4 x = (4)/(3) Jibu la mwisho: x = (4)/(3) 2. Treni ya abiria ina abiria n katika safari fulani. Kila abiria alilipa nauli ya Ksh (2n+20) na jumla ya nauli iliyokusanywa ilikuwa Ksh 120,000. Bainisha idadi ya abiria. Step 1: Unda mlinganyo unaowakilisha jumla ya nauli iliyokusanywa. Jumla ya nauli = (Idadi ya abiria) × (Nauli kwa kila abiria) 120,000 = n(2n + 20) Step 2: Panua na panga upya mlinganyo katika mfumo wa kawaida wa quadratic. 120,000 = 2n^2 + 20n 2n^2 + 20n - 120,000 = 0 Step 3: Rahisisha mlinganyo kwa kugawanya pande zote mbili kwa 2. n^2 + 10n - 60,000 = 0 Step 4: Tumia fomula ya quadratic, n = -b ± sqrt(b^2 - 4ac)2a, kutatua kwa n. Hapa, a=1, b=10, c=-60,000. n = -10 ± sqrt(10^2 - 4(1)(-60,000))2(1) n = -10 ± sqrt(100 + 240,000)2 n = -10 ± sqrt(240,100)2 n = (-10 ± 490)/(2) Step 5: Kokotoa thamani zinazowezekana za n. n_1 = (-10 + 490)/(2) = (480)/(2) = 240 n_2 = (-10 - 490)/(2) = (-500)/(2) = -250 Kwa kuwa idadi ya abiria haiwezi kuwa hasi, tunachukua thamani chanya. Jibu la mwisho: Idadi ya abiria ni 240. 3. Tafuta kituo cha ukuzaji na bainisha kipimo cha ukuzaji (linear scale factor). Step 1: Tafuta kituo cha ukuzaji. Kituo cha ukuzaji hupatikana kwa kuchora mistari iliyonyooka inayounganisha ncha zinazolingana za umbo asili (QRS) na umbo lililokuzwa (Q'R'S'). Kwa mfano, unganisha Q na Q', R na R', na S na S'. Sehemu ambapo mistari hii inakutana ndio kituo cha ukuzaji. Kutoka kwenye mchoro, sehemu hii imeandikwa kama O. Kituo cha ukuzaji ni O. Step 2: Bainisha kipimo cha ukuzaji (linear scale factor). Kipimo cha ukuzaji (k) ni uwiano wa urefu wa upande wa umbo lililokuzwa kwa urefu wa upande unaolingana wa umbo asili. Kutokana na mahesabu yaliyotolewa, tunatumia uwiano wa 8.7 kwa 3. k = Urefu wa upande wa umbo lililokuzwaUrefu wa upande wa umbo asili k = (8.7)/(3) k = 2.9 Jibu la mwisho: Kituo cha ukuzaji ni O na kipimo cha ukuzaji ni 2.9. Tuma swali linalofuata 📸

Solve 9x2 + 16 = 24x, A passenger train has n passengers on a certain trip. Each of the passengers paid a fare of Ksh (2n+20) and the total fare collected was Ksh. 120,000. Determine the number of passengers. Locate the centre of enlargement and determine the linear scale factor.

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Solve 9x2 + 16 = 24x, A passenger train has n passengers on a certain trip. Each of the passengers paid a fare of Ksh (2n+20) and the total fare collected was Ksh. 120,000. Determine the number of passengers. Locate the centre of enlargement and determine the linear scale factor.

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