Question

What must be added to x^2 - 5x to make the expression a perfect square? Express the result as the square of a bracketed expression

Answer 1

To make the expression \( x^2 - 5x \) a perfect square, we can complete the square.

1. Start with the expression:
\[
x^2 - 5x
\]

2. To complete the square, take half of the coefficient of \( x \), which is \( -5 \):
\[
\frac{-5}{2} = -\frac{5}{2}
\]

3. Square this result:
\[
\left(-\frac{5}{2}\right)^2 = \frac{25}{4}
\]

4. Thus, we need to add \( \frac{25}{4} \) to the expression \( x^2 - 5x \) to make it a perfect square:
\[
x^2 - 5x + \frac{25}{4}
\]

5. This can be factored as:
\[
\left(x - \frac{5}{2}\right)^2
\]

Therefore, the complete result is that you need to add \( \frac{25}{4} \) to \( x^2 - 5x \) to make it a perfect square, resulting in:
\[
\left(x - \frac{5}{2}\right)^2
\]