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Find the value of k if the expression 4x2 - 12x + k + 8 is a perfect square.

Mathematics

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Answer

12\frac{1}{2}

Step-by-step solution

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Ili kutafuta thamani ya kk ambapo usemi 4x2−12x+2k+84x^2 - 12x + 2k + 8 ni mraba kamili, tunahitaji kuhakikisha kuwa sehemu ya mraba kamili inafanana na (ax+b)2(ax + b)^2.

  1. Andika usemi wa mraba kamili.

    (ax+b)2=a2x2+2abx+b2(ax + b)^2 = a^2x^2 + 2abx + b^2

  2. Linganisha na 4x2−12x+2k+84x^2 - 12x + 2k + 8.

    • a2=4  ⟹  a=2a^2 = 4 \implies a = 2 (chukua thamani chanya)
    • 2ab=−12  ⟹  2⋅2⋅b=−12  ⟹  b=−32ab = -12 \implies 2 \cdot 2 \cdot b = -12 \implies b = -3

  3. Tafuta b2b^2.

    b2=(−3)2=9b^2 = (-3)^2 = 9

  4. Linganisha sehemu ya mara kwa mara.

    b2=2k+8b^2 = 2k + 8 9=2k+89 = 2k + 8

  5. Tatua kwa kk.

    2k=9−82k = 9 - 8 2k=12k = 1 k=12k = \frac{1}{2}

    Thamani ya kk ni:

    12\frac{1}{2}
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