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What must be added to the expression x2 + 8x/3 to make it a perfect square?

Mathematics

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Answer

C. 169\text{C. } \frac{16}{9}

Step-by-step solution

4 steps

To make the expression x2+8x3x^2 + \frac{8x}{3} a perfect square, we need to add a constant term. A perfect square trinomial has the form (x+b)2=x2+2bx+b2(x+b)^2 = x^2 + 2bx + b^2.

  1. Identify the coefficient of the xx term in the given expression.

    The expression is x2+83xx^2 + \frac{8}{3}x. Comparing this to x2+2bxx^2 + 2bx, we have 2b=832b = \frac{8}{3}.

  2. Solve for bb.

    2b=832b = \frac{8}{3}

    Divide both sides by 2:

    b=83×2b = \frac{8}{3 \times 2} b=86b = \frac{8}{6}

    Simplify the fraction:

    b=43b = \frac{4}{3}

  3. Calculate the constant term b2b^2 that completes the square.

    The term to be added is b2b^2.

    b2=(43)2b^2 = \left(\frac{4}{3}\right)^2 b2=4232b^2 = \frac{4^2}{3^2} b2=169b^2 = \frac{16}{9}

  4. Match the result with the given options.

    The value 169\frac{16}{9} corresponds to option C.

    The final answer is C. 169\text{C. } \frac{16}{9}.

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