Which constant term would mean that the expression is completely factored?
$x^2-3 x+$
The goal is to determine the constant term \(c\) that makes the quadratic expression \(x^2 - 3x + c\) a perfect square trinomial. A quadratic trinomial of the form \(x^2 + bx + c\) is considered a perfect square if it can be written in the form \((x + a)^2\). Expanding this binomial, we get \(x^2 + 2ax + a^2\). By comparing this with the original expression, we identify that \(b = 2a\) and \(c = a^2\). Substituting \(a = \frac{b}{2}\), we find the condition for a perfect square is \(c = \left(\frac{b}{2}\right)^2\).
In our expression \(x^2 - 3x + c\), the coefficient of the \(x\)-term is \(b = -3\). To make this a perfect square trinomial, we apply the condition derived in the first step:
\[c = \left(\frac{b}{2}\right)^2\]
Substituting \(b = -3\):
\[c = \left(\frac{-3}{2}\right)^2\]
\[c = \frac{(-3)^2}{2^2} = \frac{9}{4}\]
Thus, the constant term that completes the square is \(\frac{9}{4}\) or \(2.25\).
By setting the constant term to \(\frac{9}{4}\), the expression becomes \(x^2 - 3x + \frac{9}{4}\). We can now factor this expression as:
\[x^2 - 3x + \left(\frac{3}{2}\right)^2 = \left(x - \frac{3}{2}\right)^2\]
This confirms that the expression is now a perfect square of a linear binomial. Therefore, the required constant term is \(\boxed{\frac{9}{4}}\).
Perfect squares are essential in various fields, such as in physics when solving for the trajectory of a projectile. When you have a quadratic equation representing the height of an object over time, completing the square allows you to easily find the vertex of the parabola, which represents the maximum height reached by the object. Similarly, in economics, optimizing profit functions often involves working with quadratic forms where completing the square simplifies the process of finding the maximum or minimum value.