To transform the expression [tex]\( x^2 - 3x \)[/tex] into a perfect square trinomial, we need to complete the square. Here’s a detailed, step-by-step process:
1. Identify the coefficient of the linear term:
The given expression is [tex]\( x^2 - 3x \)[/tex]. The coefficient of the linear term [tex]\( x \)[/tex] is -3.
2. Take half of the coefficient of [tex]\( x \)[/tex] and square it:
The coefficient of [tex]\( x \)[/tex] is -3. Half of -3 is [tex]\( \frac{-3}{2} \)[/tex].
Squaring [tex]\( \frac{-3}{2} \)[/tex] gives:
[tex]\[
\left(\frac{-3}{2}\right)^2 = \frac{9}{4}
\][/tex]
3. Add this squared value to the expression to complete the square:
By adding [tex]\( \frac{9}{4} \)[/tex] to the expression [tex]\( x^2 - 3x \)[/tex], we turn it into a perfect square trinomial:
[tex]\[
x^2 - 3x + \frac{9}{4}
\][/tex]
4. Rewrite the perfect square trinomial:
This can be rewritten as:
[tex]\[
\left( x - \frac{3}{2} \right)^2
\][/tex]
Therefore, [tex]\( x^2 - 3x + \frac{9}{4} \)[/tex] is a perfect square trinomial.
Conclusion:
Thus, the value that must be added to [tex]\( x^2 - 3x \)[/tex] to make it a perfect square trinomial is [tex]\( \frac{9}{4} \)[/tex].
Therefore, the correct answer is [tex]\( \boxed{\frac{9}{4}} \)[/tex].
