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

1. Tatua mlinganyo wa quadratic

a) $X^2 + 25 = 0$

Step 1: Tenga $X^2$ upande mmoja wa mlinganyo. $X^2 = -25$

Step 2: Tafuta mzizi wa mraba wa pande zote mbili. Kumbuka kuwa $\sqrt{-1} = j$. $X = \pm \sqrt{-25}$ $X = \pm \sqrt{25 \times -1}$ $X = \pm \sqrt{25} \times \sqrt{-1}$ $X = \pm 5j$ Suluhisho ni $\boxed{X = \pm 5j}$.

b) $4t^2 - 5t + 7 = 0$

Huu ni mlinganyo wa quadratic wa fomu $at^2 + bt + c = 0$, ambapo $a=4$, $b=-5$, na $c=7$.

Step 1: Tumia fomula ya quadratic: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Step 2: Badilisha thamani za $a, b, c$ kwenye fomula. $t = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(4)(7)}}{2(4)}$

Step 3: Rahisisha usemi chini ya mzizi wa mraba. $t = \frac{5 \pm \sqrt{25 - 112}}{8}$ $t = \frac{5 \pm \sqrt{-87}}{8}$

Step 4: Andika $\sqrt{-87}$ kama $j\sqrt{87}$ (kwa kuwa $j = \sqrt{-1}$). $t = \frac{5 \pm j\sqrt{87}}{8}$ Suluhisho ni $\boxed{t = \frac{5 \pm j\sqrt{87}}{8}}$.

2. Tathmini

a) $\frac{-4}{\sqrt{9}}$

Step 1: Tafuta mzizi wa mraba wa 9. $\sqrt{9} = 3$

Step 2: Gawanya -4 kwa 3. $\frac{-4}{3}$ Thamani ni $\boxed{-\frac{4}{3}}$.

b) $j \left( \frac{1+j3}{1-j2} \right)$

Step 1: Kwanza, rahisha sehemu ya namba changamano kwa kuzidisha nambari na denominata kwa conjugate ya denominata. Conjugate ya $1-j2$ ni $1+j2$. $\frac{1+j3}{1-j2} \times \frac{1+j2}{1+j2}$

Step 2: Panua nambari na denominata. Kumbuka kuwa $j^2 = -1$. Nambari: $(1+j3)(1+j2) = 1 + j2 + j3 + j^26 = 1 + j5 - 6 = -5 + j5$ Denominata: $(1-j2)(1+j2) = 1^2 - (j2)^2 = 1 - j^24 = 1 - (-1)4 = 1 + 4 = 5$

Step 3: Badilisha nambari na denominata zilizorahisishwa kwenye sehemu. $\frac{-5 + j5}{5} = \frac{-5}{5} + \frac{j5}{5} = -1 + j$

Step 4: Sasa zidisha matokeo haya kwa $j$. $j(-1 + j) = -j + j^2$ $= -j - 1$ Thamani ni $\boxed{-1 - j}$.

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