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.
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Hizi hapa ni hatua za kutatua milinganyo ya radical: a) Tatua sqrt(4-x) - sqrt(6+x) = sqrt(14+2x) Step 1: Panga upya mlinganyo ili kurahisisha kuweka pande zote mbili kwenye mraba. sqrt(4-x) = sqrt(14+2x) + sqrt(6+x) Step 2: Weka pande zote mbili kwenye mraba. (sqrt(4-x))^2 = (sqrt(14+2x) + sqrt(6+x))^2 4-x = (14+2x) + 2sqrt((14+2x)(6+x)) + (6+x) 4-x = 20+3x + 2sqrt(84+14x+12x+2x^2) 4-x = 20+3x + 2sqrt(2x^2+26x+84) Step 3: Tenga radical iliyobaki na urahisishe. 4-x - (20+3x) = 2sqrt(2x^2+26x+84) -16-4x = 2sqrt(2x^2+26x+84) Gawanya pande zote mbili kwa 2: -8-2x = sqrt(2x^2+26x+84) Step 4: Weka pande zote mbili kwenye mraba tena. (-8-2x)^2 = (sqrt(2x^2+26x+84))^2 (2x+8)^2 = 2x^2+26x+84 4x^2+32x+64 = 2x^2+26x+84 Step 5: Panga upya kuwa mlinganyo wa quadratic. 4x^2-2x^2+32x-26x+64-84 = 0 2x^2+6x-20 = 0 Gawanya kwa 2: x^2+3x-10 = 0 Step 6: Tatua mlinganyo wa quadratic. (x+5)(x-2) = 0 Hivyo, x=-5 au x=2. Angalia majibu katika mlinganyo asili. Kwa x=-5: sqrt(4-(-5)) - sqrt(6+(-5)) = sqrt(9) - sqrt(1) = 3-1=2. Na sqrt(14+2(-5)) = sqrt(14-10) = sqrt(4)=2. Hivyo x=-5 ni suluhisho sahihi. Kwa x=2: sqrt(4-2) - sqrt(6+2) = sqrt(2) - sqrt(8) = sqrt(2) - 2sqrt(2) = -sqrt(2). Na sqrt(14+2(2)) = sqrt(18) = 3sqrt(2). Kwa kuwa -sqrt(2) ≠ 3sqrt(2), x=2 si suluhisho sahihi. Suluhisho la mlinganyo ni: x=-5 b) Tatua 2sqrt(2x-12) + sqrt(2x-3) - 3 = 0 Step 1: Tenga radical moja. 2sqrt(2x-12) = 3 - sqrt(2x-3) Step 2: Weka pande zote mbili kwenye mraba. (2sqrt(2x-12))^2 = (3 - sqrt(2x-3))^2 4(2x-12) = 3^2 - 2(3)sqrt(2x-3) + (sqrt(2x-3))^2 8x-48 = 9 - 6sqrt(2x-3) + 2x-3 8x-48 = 6+2x - 6sqrt(2x-3) Step 3: Tenga radical iliyobaki. 8x-48 - (6+2x) = -6sqrt(2x-3) 6x-54 = -6sqrt(2x-3) Step 4: Gawanya pande zote mbili kwa -6. (6x-54)/(-6) = sqrt(2x-3) -x+9 = sqrt(2x-3) Step 5: Weka pande zote mbili kwenye mraba tena. (-x+9)^2 = (sqrt(2x-3))^2 x^2-18x+81 = 2x-3 Step 6: Panga upya kuwa mlinganyo wa quadratic. x^2-18x-2x+81+3 = 0 x^2-20x+84 = 0 Step 7: Tatua mlinganyo wa quadratic kwa kutumia formula ya quadratic x = -b ± sqrt(b^2-4ac)2a. x = -(-20) ± sqrt((-20)^2 - 4(1)(84))2(1) x = 20 ± sqrt(400 - 336)2 x = 20 ± sqrt(64)2 x = (20 ± 8)/(2) Hivyo, x_1 = (20+8)/(2) = (28)/(2) = 14 au x_2 = (20-8)/(2) = (12)/(2) = 6. Step 8: Angalia majibu katika mlinganyo asili. Kwa x=14: 2sqrt(2(14)-12) + sqrt(2(14)-3) - 3 = 2sqrt(16) + sqrt(25) - 3 = 2(4) + 5 - 3 = 8+5-3 = 10. Kwa kuwa 10 ≠ 0, x=14 si suluhisho sahihi. Kwa x=6: 2sqrt(2(6)-12) + sqrt(2(6)-3) - 3 = 2sqrt(0) + sqrt(9) - 3 = 2(0) + 3 - 3 = 0. Hivyo x=6 ni suluhisho sahihi. Suluhisho la mlinganyo ni: x=6 Tuma swali linalofuata 📸