And the other questions 14-16?

Mathematics

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2 parts
15

x=3 au x=−12x = 3 \text{ au } x = -\frac{1}{2}

16

x=5 au x=1x = 5 \text{ au } x = 1

Step-by-step solution

11 steps

Step 1: Gawanya pande zote mbili za mlinganyo kwa mgawo wa x2x^2, ambao ni 3.

3x2−10x+8=03x^2 - 10x + 8 = 0 3x23−10x3+83=03\frac{3x^2}{3} - \frac{10x}{3} + \frac{8}{3} = \frac{0}{3} x2−103x+83=0x^2 - \frac{10}{3}x + \frac{8}{3} = 0

Step 2: Hamisha namba isiyo na xx (constant term) upande wa kulia wa mlinganyo.

x2−103x=−83x^2 - \frac{10}{3}x = -\frac{8}{3}

Step 3: Kamilisha mraba upande wa kushoto. Ongeza (12×mgawo wa x)2(\frac{1}{2} \times \text{mgawo wa } x)^2 pande zote mbili. Mgawo wa xx ni −103-\frac{10}{3}.

(12×−103)2=(−53)2=259\left(\frac{1}{2} \times -\frac{10}{3}\right)^2 = \left(-\frac{5}{3}\right)^2 = \frac{25}{9} x2−103x+259=−83+259x^2 - \frac{10}{3}x + \frac{25}{9} = -\frac{8}{3} + \frac{25}{9}

Step 4: Rahisisha upande wa kulia na andika upande wa kushoto kama mraba kamili.

x2−103x+259=−249+259x^2 - \frac{10}{3}x + \frac{25}{9} = -\frac{24}{9} + \frac{25}{9} (x−53)2=19\left(x - \frac{5}{3}\right)^2 = \frac{1}{9}

Step 5: Chukua mzizi wa mraba pande zote mbili.

(x−53)2=±19\sqrt{\left(x - \frac{5}{3}\right)^2} = \pm\sqrt{\frac{1}{9}} x−53=±13x - \frac{5}{3} = \pm\frac{1}{3}

Step 6: Tatua kwa xx.

x=53±13x = \frac{5}{3} \pm \frac{1}{3}

Kuna suluhisho mbili:

x1=53+13=63=2x_1 = \frac{5}{3} + \frac{1}{3} = \frac{6}{3} = 2 x2=53−13=43x_2 = \frac{5}{3} - \frac{1}{3} = \frac{4}{3}

Majibu ni:

x=2 au x=43\boxed{x = 2 \text{ au } x = \frac{4}{3}}

Part 15· 6 steps

Tatua mlinganyo 2x2−5x−3=02x^2 - 5x - 3 = 0 kwa kukamilisha mraba.

  1. Gawanya pande zote mbili za mlinganyo kwa mgawo wa x2x^2, ambao ni 2.

    2x2−5x−3=02x^2 - 5x - 3 = 0 2x22−5x2−32=02\frac{2x^2}{2} - \frac{5x}{2} - \frac{3}{2} = \frac{0}{2} x2−52x−32=0x^2 - \frac{5}{2}x - \frac{3}{2} = 0

  2. Hamisha namba isiyo na xx (constant term) upande wa kulia wa mlinganyo.

    x2−52x=32x^2 - \frac{5}{2}x = \frac{3}{2}

  3. Kamilisha mraba upande wa kushoto.

    Ongeza (12×mgawo wa x)2(\frac{1}{2} \times \text{mgawo wa } x)^2 pande zote mbili. Mgawo wa xx ni −52-\frac{5}{2}.

    (12×−52)2=(−54)2=2516\left(\frac{1}{2} \times -\frac{5}{2}\right)^2 = \left(-\frac{5}{4}\right)^2 = \frac{25}{16} x2−52x+2516=32+2516x^2 - \frac{5}{2}x + \frac{25}{16} = \frac{3}{2} + \frac{25}{16}

  4. Rahisisha upande wa kulia na andika upande wa kushoto kama mraba kamili.

    x2−52x+2516=2416+2516x^2 - \frac{5}{2}x + \frac{25}{16} = \frac{24}{16} + \frac{25}{16} (x−54)2=4916\left(x - \frac{5}{4}\right)^2 = \frac{49}{16}

  5. Chukua mzizi wa mraba pande zote mbili.

    (x−54)2=±4916\sqrt{\left(x - \frac{5}{4}\right)^2} = \pm\sqrt{\frac{49}{16}} x−54=±74x - \frac{5}{4} = \pm\frac{7}{4}

  6. Tatua kwa xx.

    x=54±74x = \frac{5}{4} \pm \frac{7}{4}

    Kuna suluhisho mbili:

    x1=54+74=124=3x_1 = \frac{5}{4} + \frac{7}{4} = \frac{12}{4} = 3 x2=54−74=−24=−12x_2 = \frac{5}{4} - \frac{7}{4} = -\frac{2}{4} = -\frac{1}{2}

    Majibu ni:

    x=3 au x=−12\boxed{x = 3 \text{ au } x = -\frac{1}{2}}

Part 16

x=5 au x=1x = 5 \text{ au } x = 1

Tatua mlinganyo x2−6x+5=0x^2 - 6x + 5 = 0 kwa kukamilisha mraba.

  1. Hamisha namba isiyo na xx (constant term) upande wa kulia wa mlinganyo.

    x2−6x=−5x^2 - 6x = -5

  2. Kamilisha mraba upande wa kushoto.

    Ongeza (12×mgawo wa x)2(\frac{1}{2} \times \text{mgawo wa } x)^2 pande zote mbili. Mgawo wa xx ni −6-6.

    (12×−6)2=(−3)2=9\left(\frac{1}{2} \times -6\right)^2 = (-3)^2 = 9 x2−6x+9=−5+9x^2 - 6x + 9 = -5 + 9

  3. Rahisisha upande wa kulia na andika upande wa kushoto kama mraba kamili.

    (x−3)2=4\left(x - 3\right)^2 = 4

  4. Chukua mzizi wa mraba pande zote mbili.

    (x−3)2=±4\sqrt{\left(x - 3\right)^2} = \pm\sqrt{4} x−3=±2x - 3 = \pm 2

  5. Tatua kwa xx.

    x=3±2x = 3 \pm 2

    Kuna suluhisho mbili:

    x1=3+2=5x_1 = 3 + 2 = 5 x2=3−2=1x_2 = 3 - 2 = 1

    Majibu ni:

    x=5 au x=1\boxed{x = 5 \text{ au } x = 1}
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