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 kwa kutumia sheria za eksponenti. Mlinganyo uliotolewa ni 7^2x - 8 × 7^x + 7 = 0. Tunaweza kuandika 7^2x kama (7^x)^2. Hivyo, mlinganyo unakuwa: (7^x)^2 - 8(7^x) + 7 = 0 Step 2: Tumia ubadilishaji ili kurahisisha mlinganyo. Acha y = 7^x. Kubadilisha y kwenye mlinganyo kunatoa mlinganyo wa quadratic: y^2 - 8y + 7 = 0 Step 3: Tatua mlinganyo wa quadratic kwa y. Mlinganyo y^2 - 8y + 7 = 0 unaweza kutatuliwa kwa kuweka mambo. Tunatafuta nambari mbili zinazozidisha na kutoa 7 na kujumlisha na kutoa -8. Nambari hizo ni -1 na -7. (y - 1)(y - 7) = 0 Hii inatoa suluhisho mbili kwa y: y - 1 = 0 y = 1 y - 7 = 0 y = 7 Step 4: Badilisha y nyuma ili kupata x. Kesi 1: y = 1 Kwa kuwa y = 7^x, tuna: 7^x = 1 Tunajua kwamba nambari yoyote isiyo sifuri ikipandishwa kwa nguvu ya 0 ni 1. Hivyo, 1 = 7^0. 7^x = 7^0 Kulinganisha eksponenti kunatoa: x = 0 Kesi 2: y = 7 Kwa kuwa y = 7^x, tuna: 7^x = 7 Hii inaweza kuandikwa kama 7^x = 7^1. Kulinganisha eksponenti kunatoa: x = 1 Step 5: Andika majibu ya mwisho. Thamani za x zinazokidhi mlinganyo ni 0 na 1. x = 0, 1 Tuma swali linalofuata 📸