3 What must be added to \( x^{2}-5 x \) to make the expestion a pertect square? Express rewult as the square of a bracketed expression The coefficient of \( x \) is \( \square \) Suppose \( x^{2}-5 x+k \) is a perfect square and thet it is equal to \( (x-a)^{2} \). \[ \begin{aligned} \Rightarrow \quad & x^{2}-5 x+k=(x-a)^{2} \\ & x^{2}-5 x+k=x^{2}-\square x+\square \end{aligned} \] * By comparing the coefficients of \( x \), \[ \begin{array}{c} -5=\square a \\ a=\square \end{array} \] 3y comparing the constant terms, \[ \begin{array}{l} k=\square \\ k=\square \\ k=\frac{\square}{2} \end{array} \] Therefore, \( \square \) must be added to the expression.

To determine what must be added to \( x^{2} - 5x \) to make it a perfect square, we can follow these steps: 1. **Identify the expression**: We have \( x^{2} - 5x \). 2. **Set up the perfect square**: We want to express \( x^{2} - 5x + k \) as a perfect square, which can be written in the form \( (x - a)^{2} \). 3. **Expand the perfect square**: \[ (x - a)^{2} = x^{2} - 2ax + a^{2} \] 4. **Equate the two expressions**: \[ x^{2} - 5x + k = x^{2} - 2ax + a^{2} \] 5. **Compare coefficients**: - For the coefficient of \( x \): \[ -5 = -2a \implies a = \frac{5}{2} \] - For the constant terms: \[ k = a^{2} = \left(\frac{5}{2}\right)^{2} = \frac{25}{4} \] 6. **Conclusion**: Therefore, to make \( x^{2} - 5x \) a perfect square, we need to add \( k = \frac{25}{4} \). 7. **Final expression**: The expression becomes: \[ x^{2} - 5x + \frac{25}{4} = \left(x - \frac{5}{2}\right)^{2} \] Now, let's summarize the results in the required format: - The coefficient of \( x \) is \( \frac{5}{2} \). - The value of \( k \) that must be added is \( \frac{25}{4} \). Thus, the final answer is: - Therefore, \( \frac{25}{4} \) must be added to the expression.