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Which value must be added to the expression [tex]$x^2 - 3x$[/tex] to make it a perfect-square trinomial?

A. [tex]\frac{3}{2}[/tex]
B. [tex]\frac{9}{4}[/tex]
C. 6
D. 9

Sagot :

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.

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].