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.)

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 \(x\).

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.

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

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

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

Final Answer