Solve the quadratic equation: (a) x2 + 25 = 0, 4t2 - 5t + 7 = 0. 2. Evaluate (a) −4 / sqrt(9), (b) j( (1+j3)…

Question

Solve the quadratic equation: (a) x2 + 25 = 0, 4t2 - 5t + 7 = 0. 2. Evaluate (a) −4 / sqrt(9), (b) j( (1+j3) / (1-j2) ).

ScanSolve AI Tutor · Accepted Answer

1. Tatua mlinganyo wa quadratic a) X² + 25 = 0 Step 1: Tenga X² upande mmoja wa mlinganyo. X² = −25; Step 2: Tafuta mzizi wa mraba wa pande zote mbili. Kumbuka kuwa √(−1) = j. X = ± √(−25); X = ± √(25 × −1); X = ± √(25) × √(−1); X = ± 5j; Suluhisho ni X = ± 5j. b) 4t² − 5t + 7 = 0; Huu ni mlinganyo wa quadratic wa fomu at² + bt + c = 0, ambapo a=4, b=−5, na c=7. Step 1: Tumia fomula ya quadratic: t = (−b ± √(b² − 4ac))/(2a). Step 2: Badilisha thamani za a, b, c kwenye fomula. t = (−(−5) ± √((−5)² − 4(4)(7)))/(2(4)) Step 3: Rahisisha usemi chini ya mzizi wa mraba. t = (5 ± √(25 − 112))/(8); t = (5 ± √(−87))/(8); Step 4: Andika √(−87) kama j√(87) (kwa kuwa j = √(−1)). t = (5 ± j√(87))/(8); Suluhisho ni t = (5 ± j√(87))/(8). 2. Tathmini a) (−4)/(√(9)) Step 1: Tafuta mzizi wa mraba wa 9. √(9) = 3 Step 2: Gawanya −4 kwa 3. (−4)/(3) Thamani ni −4/3. b) j ( (1+j3)/(1−j2) ) Step 1: Kwanza, rahisha sehemu ya namba changamano kwa kuzidisha nambari na denominata kwa conjugate ya denominata. Conjugate ya 1−j2 ni 1+j2. (1+j3)/(1−j2) × (1+j2)/(1+j2) Step 2: Panua nambari na denominata. Kumbuka kuwa j² = −1. Nambari: (1+j3)(1+j2) = 1 + j2 + j3 + j²⁶ = 1 + j5 − 6 = −5 + j5; Denominata: (1−j2)(1+j2) = 1² − (j2)² = 1 − j²⁴ = 1 − (−1)4 = 1 + 4 = 5 Step 3: Badilisha nambari na denominata zilizorahisishwa kwenye sehemu. (−5 + j5)/(5) = (−5)/(5) + j5/5 = −1 + j Step 4: Sasa zidisha matokeo haya kwa j. j(−1 + j) = −j + j²; = −j − 1 Thamani ni −1 − j.

Solve the quadratic equation: (a) x2 + 25 = 0, 4t2 − 5t + 7 = 0. 2. Evaluate (a) -4 / sqrt(9), (b) j( (1+j3) / (1-j2) ).

Step-by-step solution

Tenga $X^2$ upande mmoja wa mlinganyo.

Tafuta mzizi wa mraba wa pande zote mbili.

Tumia fomula ya quadratic

Badilisha thamani za $a, b, c$ kwenye fomula.

Rahisisha usemi chini ya mzizi wa mraba.

Andika $\sqrt{-87}$ kama $j\sqrt{87}$ (kwa kuwa $j = \sqrt{-1}$ ).

Tafuta mzizi wa mraba wa 9.

Gawanya -4 kwa 3.

Kwanza, rahisha sehemu ya namba changamano kwa kuzidisha nambari na denominata kwa conjugate ya denominata.

Panua nambari na denominata.

Badilisha nambari na denominata zilizorahisishwa kwenye sehemu.

Sasa zidisha matokeo haya kwa $j$ .

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