Questions: Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then, write and factor the trinomial. x^2 - (4/5) x What is the constant that should be added to the binomial so that it becomes a perfect square trinomial? (Type a simplified fraction.)

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then, write and factor the trinomial. x^2 - (4/5) x What is the constant that should be added to the binomial so that it becomes a perfect square trinomial? (Type a simplified fraction.)
Transcript text: Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then, write and factor the trinomial. \[ x^{2}-\frac{4}{5} x \] What is the constant that should be added to the binomial so that it becomes a perfect square trinomial? $\square$ (Type a simplified fraction.)
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Solution

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Solution Steps

To determine the constant that should be added to the binomial \( x^2 - \frac{4}{5} x \) so that it becomes a perfect square trinomial, we need to use the formula for completing the square. The constant to be added is \(\left(\frac{b}{2}\right)^2\), where \(b\) is the coefficient of \(x\).

Solution Approach
  1. Identify the coefficient of \(x\), which is \(-\frac{4}{5}\).
  2. Divide this coefficient by 2.
  3. Square the result to find the constant that should be added.
Step 1: Identify the Coefficient of \(x\)

The given binomial is \(x^2 - \frac{4}{5} x\). The coefficient of \(x\) is \(-\frac{4}{5}\).

Step 2: Divide the Coefficient by 2

To complete the square, we need to divide the coefficient of \(x\) by 2: \[ \frac{-\frac{4}{5}}{2} = -\frac{2}{5} \]

Step 3: Square the Result

Next, we square the result from Step 2: \[ \left(-\frac{2}{5}\right)^2 = \frac{4}{25} \approx 0.1600 \]

Final Answer

The constant that should be added to the binomial \(x^2 - \frac{4}{5} x\) to make it a perfect square trinomial is: \[ \boxed{\frac{4}{25}} \]

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