Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. \( x^{2}-\frac{4}{5} x \)

To determine the constant that should be added to the binomial \( x^{2} - \frac{4}{5}x \) to make it a perfect square trinomial, follow these steps: ### Step 1: Understand the Form of a Perfect Square Trinomial A perfect square trinomial has the form: \[ (x + a)^{2} = x^{2} + 2ax + a^{2} \] Comparing this with the given binomial \( x^{2} - \frac{4}{5}x \), we can see that: \[ 2a = -\frac{4}{5} \] ### Step 2: Solve for \( a \) \[ a = \frac{-\frac{4}{5}}{2} = -\frac{2}{5} \] ### Step 3: Find the Constant to Add The constant to add is \( a^{2} \): \[ a^{2} = \left(-\frac{2}{5}\right)^{2} = \frac{4}{25} \] ### Conclusion To make \( x^{2} - \frac{4}{5}x \) a perfect square trinomial, you should add \( \frac{4}{25} \). **Answer:** The constant to add is four twenty-fifths, that is \( \frac{4}{25} \).