To complete the expression so it forms a perfect-square trinomial, we need to follow the process of completing the square.
## Part 1: Completing [tex]\( x^2 - 5x + \cdots \)[/tex]
1. Start with the given quadratic expression: [tex]\( x^2 - 5x \)[/tex].
2. Take the coefficient of [tex]\( x \)[/tex], which is [tex]\( -5 \)[/tex].
3. Divide this coefficient by 2: [tex]\( \frac{-5}{2} = -\frac{5}{2} \)[/tex].
4. Square the result from step 3: [tex]\((-\frac{5}{2})^2 = \frac{25}{4} \)[/tex].
5. Add this square to the quadratic expression to form a perfect-square trinomial: [tex]\( x^2 - 5x + \frac{25}{4} \)[/tex].
So, the completed expression is [tex]\( x^2 - 5x + \frac{25}{4} \)[/tex].
## Part 2: Completing [tex]\( x^2 + \square x + 49 \)[/tex]
Next, we need to complete the square for [tex]\( x^2 + kx + 49 \)[/tex] by finding the appropriate coefficient [tex]\( k \)[/tex]:
1. Let’s denote the coefficient of [tex]\( x \)[/tex] by [tex]\( k \)[/tex].
2. Since the constant term [tex]\( 49 \)[/tex] is a perfect square, this term is equivalent to [tex]\( 7^2 \)[/tex].
3. To form a perfect square trinomial of the form [tex]\( (x + c)^2 \)[/tex], [tex]\( c \)[/tex] would need to be added and then squared to yield the constant.
4. Set the equation for [tex]\( c \)[/tex]: [tex]\( \left(\frac{k}{2}\right)^2 = 49 \)[/tex].
5. Solving for [tex]\( k \)[/tex]:
- [tex]\(\left(\frac{k}{2}\right)^2 = 49\)[/tex],
- [tex]\(\frac{k^2}{4} = 49\)[/tex],
- [tex]\( k^2 = 196 \)[/tex],
- [tex]\( k = \pm 14 \)[/tex].
So the coefficient [tex]\( k \)[/tex] found is [tex]\( \pm 14 \)[/tex].
Therefore, the completed perfect-square trinomial is either [tex]\( x^2 + 14x + 49 \)[/tex] or [tex]\( x^2 - 14x + 49 \)[/tex].
### Final Answers:
1. [tex]\( x^2 - 5x + \frac{25}{4} \)[/tex]
2. [tex]\( x^2 + 14x + 49 \)[/tex] or [tex]\( x^2 - 14x + 49 \)[/tex]
