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Solve the equation using the completing square method. 28x2 + 3x − 1 = 0

Mathematics

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Answer

x=17 au x=−14x = \frac{1}{7} \text{ au } x = -\frac{1}{4}

Step-by-step solution

6 steps

  1. Gawanya pande zote mbili za mlinganyo kwa mgawo wa x2x^2, ambao ni 28.

    28x2+3x−1=028x^2 + 3x - 1 = 0 28x228+3x28−128=028\frac{28x^2}{28} + \frac{3x}{28} - \frac{1}{28} = \frac{0}{28} x2+328x−128=0x^2 + \frac{3}{28}x - \frac{1}{28} = 0

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

    x2+328x=128x^2 + \frac{3}{28}x = \frac{1}{28}

  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 328\frac{3}{28}.

    (12×328)2=(356)2=93136\left(\frac{1}{2} \times \frac{3}{28}\right)^2 = \left(\frac{3}{56}\right)^2 = \frac{9}{3136} x2+328x+93136=128+93136x^2 + \frac{3}{28}x + \frac{9}{3136} = \frac{1}{28} + \frac{9}{3136}

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

    x2+328x+93136=1123136+93136x^2 + \frac{3}{28}x + \frac{9}{3136} = \frac{112}{3136} + \frac{9}{3136} (x+356)2=1213136\left(x + \frac{3}{56}\right)^2 = \frac{121}{3136}

  5. Chukua mzizi wa mraba pande zote mbili.

    (x+356)2=±1213136\sqrt{\left(x + \frac{3}{56}\right)^2} = \pm\sqrt{\frac{121}{3136}} x+356=±1156x + \frac{3}{56} = \pm\frac{11}{56}

  6. Tatua kwa xx.

    x=−356±1156x = -\frac{3}{56} \pm \frac{11}{56}

    Kuna suluhisho mbili:

    x1=−356+1156=856=17x_1 = -\frac{3}{56} + \frac{11}{56} = \frac{8}{56} = \frac{1}{7} x2=−356−1156=−1456=−14x_2 = -\frac{3}{56} - \frac{11}{56} = -\frac{14}{56} = -\frac{1}{4}

    Majibu ni:

    x=17 au x=−14x = \frac{1}{7} \text{ au } x = -\frac{1}{4}
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