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Calculate: Use completing of the square method to solve 2x2 + 7x − 9 = 0

Mathematics

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Answer

x=1 au x=−92x = 1 \text{ au } x = -\frac{9}{2}

Step-by-step solution

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Huu hapa utatuzi wa mlinganyo 2x2+7x−9=02x^2 + 7x - 9 = 0 kwa kutumia njia ya kukamilisha mraba:

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

    2x22+7x2−92=02\frac{2x^2}{2} + \frac{7x}{2} - \frac{9}{2} = \frac{0}{2} x2+72x−92=0x^2 + \frac{7}{2}x - \frac{9}{2} = 0

  2. Hamisha nambari isiyobadilika (constant term) kwenda upande wa kulia wa mlinganyo.

    x2+72x=92x^2 + \frac{7}{2}x = \frac{9}{2}

  3. Kamilisha mraba kwa kuongeza (b2)2(\frac{b}{2})^2 kwa pande zote mbili.

    Katika mlinganyo huu, b=72b = \frac{7}{2}.

    (b2)2=(7/22)2=(74)2=4916\left(\frac{b}{2}\right)^2 = \left(\frac{7/2}{2}\right)^2 = \left(\frac{7}{4}\right)^2 = \frac{49}{16}

    Ongeza 4916\frac{49}{16} kwa pande zote mbili:

    x2+72x+4916=92+4916x^2 + \frac{7}{2}x + \frac{49}{16} = \frac{9}{2} + \frac{49}{16}

  4. Andika upande wa kushoto kama mraba kamili na rahisha upande wa kulia.

    (x+74)2=7216+4916\left(x + \frac{7}{4}\right)^2 = \frac{72}{16} + \frac{49}{16} (x+74)2=12116\left(x + \frac{7}{4}\right)^2 = \frac{121}{16}

  5. Chukua mzizi wa mraba wa pande zote mbili.

    (x+74)2=±12116\sqrt{\left(x + \frac{7}{4}\right)^2} = \pm\sqrt{\frac{121}{16}} x+74=±114x + \frac{7}{4} = \pm\frac{11}{4}

  6. Tatua kwa xx kwa kutenganisha kesi mbili (chanya na hasi).

    Kesi ya kwanza:

    x+74=114x + \frac{7}{4} = \frac{11}{4} x=114−74x = \frac{11}{4} - \frac{7}{4} x=44x = \frac{4}{4} x=1x = 1

    Kesi ya pili:

    x+74=−114x + \frac{7}{4} = -\frac{11}{4} x=−114−74x = -\frac{11}{4} - \frac{7}{4} x=−184x = -\frac{18}{4} x=−92x = -\frac{9}{2}

    Majibu ni:

    x=1 au x=−92x = 1 \text{ au } x = -\frac{9}{2}
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